We present single-crystal neutron scattering measurements of the spin-1/2 equilateral triangular-lattice antiferromagnet Ba3CoSb2O9. Besides confirming that the Co 2+ magnetic moments lie in the ab plane for zero magnetic field and then determining all the exchange parameters of the minimal quasi-2D spin Hamiltonian, we provide conclusive experimental evidence of magnon decay through observation of intrinsic line-broadening. Through detailed comparisons with the linear and nonlinear spin-wave theories, we also point out that the large-S approximation, which is conventionally employed to predict magnon decay in noncollinear magnets, is inadequate to explain our experimental observation. Thus, our results call for a new theoretical framework for describing excitation spectra in low-dimensional frustrated magnets under strong quantum effects. The equilateral triangular-lattice quantum antiferromagnet Ba 3 CoSb 2 O 9 was synthesized recently [24][25][26][27][28][29]. The Co 2+ ion has a Kramers doublet ground-state due to the spin-orbit coupling, and this doublet can be described as an effective spin-1/2 moment. In addition, the high symmetry of the hexagonal crystal structure, P6 3 / mmc [24-28], forbids DM interaction for pairs up to third nearest-neighbor (NN) in the same abplane and between any pair of spins along the c-axis [25].Powder neutron diffraction measurements presented the noncollinear 120• structure with the magnetic wavevector Q = (1/3, 1/3, 1) [24]. The Néel temperature was found to be ≈ 3.8 K and a rich temperature-magnetic field phase diagram was reported up to 32 T [25][26][27][28]. Electronic spin resonance (ESR) [27] and nuclear magnetic resonance (NMR) [28] measurements suggested a spin model with small easy-plane exchange anisotropy and an exchange interaction along the caxis much weaker than the NN intralayer exchange. This observation is consistent with the alternation of magnetic (Co 2+ ) and nonmagnetic (Sb 2 O 9 bioctahedra) layers along the cdirection. While more precise determination of the model parameters requires inelastic neutron scattering (INS) measurements, such detailed information is indeed physically relevant.

The term frustration refers to lattice systems whose ground state cannot simultaneously satisfy all the interactions. Frustration is an important property of correlated electron systems, which stems from the sign of loop products (similar to Wilson products) of interactions on a lattice. It was early recognized that geometric frustration can produce rather exotic physical behaviors, such as macroscopic ground state degeneracy and helimagnetism. The interest in frustrated systems was renewed two decades later in the context of spin glasses and the emergence of magnetic superstructures. In particular, Phil Anderson's proposal of a quantum spin liquid ground state for a two-dimensional lattice S  =  1/2 Heisenberg magnet generated a very active line of research that still continues. As a result of these early discoveries and conjectures, the study of frustrated models and materials exploded over the last two decades. Besides the large efforts triggered by the search of quantum spin liquids, it was also recognized that frustration plays a crucial role in a vast spectrum of physical phenomena arising from correlated electron materials. Here we review some of these phenomena with particular emphasis on the stabilization of chiral liquids and non-coplanar magnetic orderings. In particular, we focus on the ubiquitous interplay between magnetic and charge degrees of freedom in frustrated correlated electron systems and on the role of anisotropy. We demonstrate that these basic ingredients lead to exotic phenomena, such as, charge effects in Mott insulators, the stabilization of single magnetic vortices, as well as vortex and skyrmion crystals, and the emergence of different types of chiral liquids. In particular, these orderings appear more naturally in itinerant magnets with the potential of inducing a very large anomalous Hall effect.

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The magnetic phases of the ideal spin-1/2 triangular-lattice antiferromagnet Ba3CoSb2O9 are identified and studied using 135,137 Ba nuclear magnetic resonance (NMR) spectroscopy in magnetic fields ranging to 30T, oriented parallel and near perpendicular to the crystallographic ab-plane. For both directions, the saturation field is approximately 33T. Notably, the NMR spectra provide microscopic evidence for the stabilization of an up-up-down spin configuration for in-plane fields, giving rise to an one-third magnetization plateau (Msat/3), as well as for a higher field phase transition near to ∼ (3/5)Msat for both field orientations. Phase transitions are signaled by the evolution of the NMR spectra, and in some cases through spin-lattice relaxation measurements. The results are compared with expectations obtained from a semi-classical energy density modeling, in which quantum effects are incorporated by effective interactions extracted from the spin-wave analysis of the two-dimensional model. The interlayer coupling also plays a significant role in the outcome. Good agreement between the model and the experimental results is achieved, except for the case of fields approaching the saturation value applied along the c-axis.

We study quantum phase transitions in the honeycomb Kitaev model under a magnetic field, focusing on the topological nature of Majorana fermion excitations. We find a gapless phase between the low-field gapless quantum spin liquid and the high-field gapped forced-ferromagnetic state for the antiferromagnetic Kitaev model in the [001] field by using the Majorana mean-field theory, in conjunction with the exact diagonalization and the spin-wave theory supporting the validity of this approach. The transition between the two gapless phases is driven by a topological change of the Majorana spectrum -line node formation interconnecting two Majorana cones. The peculiar change of the Majorana band topology is rationalized by a sign change of the effective Kitaev coupling by the magnetic field, which does not occur in the ferromagnetic Kitaev case. Upon tilting the magnetic field away from [001], the two gapless phases become gapped and topologically nontrivial, characterized by nonzero Chern numbers with different signs. The sign change of the Chern number leads to a reversal of the thermal edge current in the half-quantized thermal Hall effect.The concept of topology plays a central role in the current forefront of condensed matter physics. This holds particularly true in the study of quantum spin liquids (QSLs), where any kind of conventional order is suppressed by quantum fluctuations [1,2]. Originating from their topological order, QSLs can host fractionalized excitations [3], which may be observed as excitation continua in dynamical spin responses and lowtemperature asymptotic behaviors of specific heat and thermal conductivity [4][5][6][7][8]. In addition, fractionalized excitations can form a nontrivial topological band structure. To observe this, thermal Hall measurements have been performed in QSL candidate materials in an applied magnetic field [8][9][10].Such topological nature and fractionalized excitations have been actively debated in the past decade for Kitaev-type QSLs. The Kitaev model has an exact QSL ground state associated with Majorana fermions emergent from spin fractionalization [11][12][13]. Although most of the candidate materials exhibit a magnetic order at low temperature [14-20], their excitation spectra observed by neutron and Raman scatterings [21][22][23][24] and longitudinal thermal conductivity [25] above Néel temperature show anomalous features likely related to Majorana fermions.Recently, the magnetic-field effect suppressing the magnetic order became a topical issue in the Kitaev candidate materials [26][27][28][29][30][31][32][33][34][35][36][37]. Theoretically, a weak magnetic field in the perturbative regime is known to open a gap in the Majorana fermion spectrum, which induces the half-quantized thermal Hall conductivity due to the Majorana chiral edge mode [11,38]. Interestingly, a thermal Hall effect was observed in the Kitaev candidate material α-RuCl 3 in a magnetic field above the Néel temperature [9, 10]. However, a magnetic field beyond the perturbative treatment violates th...

We study a frustrated quantum Ising model relevant for Ca 3 Co 2 O 6 that consists of a triangular lattice of weakly-coupled ferromagnetic (FM) chains. According to our quantum Monte Carlo (QMC) simulations, the chains become FM and form a three-sublattice "up-up-down" structure for T ≤ T CI . In contrast, long-period spin-density-wave (SDW) microphases are stabilized along the chains for T CI < T < T c . Our mean field solutions reveal a quasi-continuous temperature dependence of the SDW wavelength, implying the existence of metastable states that explain the very slow dynamics observed in Ca 3 Co 2 O 6 . We also discuss implications of microphases for the related multiferroic compounds Ca 3 CoMnO 6 and Lu 2 MnCoO 6 . [3][4][5], the spin-driven "nematic" transition in pnictides [6][7][8][9], and dimensional reduction in BaCuSi 2 O 6 [10,11]. Ca 3 Co 2 O 6 is another example comprising a triangular lattice of FM Ising chains coupled by weak antiferromagnetic (AFM) exchanges. This compound exhibits field-induced magnetization steps whose heights depend on the field sweep history and rate [12][13][14][15][16]. We will show that this out-of-equilibrium behavior has its roots in exotic equilibrium properties that can be extended to the related multiferroic compound Ca 3 Co 2−x Mn x O 6 [17][18][19][20][21][22].The Co 3+ ions (Co II) on the trigonal prism sites of Ca 3 Co 2 O 6 contain 3d 6 localized electrons that generate an S = 2 spin with large Ising-like anisotropy [23][24][25][26]. These ions form a triangular lattice of FM Ising chains along the caxis (Fig. 1) and the structure comprises three sublattices of layers stacked along the c-axis in an ABCABC . . . configuration. Although the AFM inter-chain couplings J 2 and J 3 [27] [ Fig. 1(a)] are an order of magnitude smaller than the intrachain FM exchange, |J 1 | = 2 × 10 K [23, 27], we will show that they strongly affect the intra-chain spin correlations over a window of temperatures below T c .The initial interest in Ca 3 Co 2 O 6 was triggered by the observation of out-of-equilibrium magnetization steps measured below ∼ 8 K and ∼ 3.6 T that appear at regular field intervals. Previous works invoked a "rigid-chain model": every chain is replaced by a single Ising spin by assuming [28][29][30][31][32]. Each spin of the resulting triangular lattice Ising model (TLIM), represents the magnetization of the whole chain and it is flipped if gµ B H overcomes its molecular field. Within this simplified framework, the regular field intervals result from the equally-spaced discrete molecular field spectrum [28]. However, this 2D scenario was challenged by the recent discovery of long-wavelength intra-chain spindensity-wave (SDW) ordering below T c 25 K [16,33,34]. Motivated by this discovery, Chapon initiated the study of a more realistic 3D lattice model by using a random-phase approximation (RPA) which is only valid close to T = T c [35].By combining QMC simulations and mean field (MF) solu-

Quantum fluctuations become particularly relevant in highly frustrated quantum magnets and can lead to new states of matter. We provide a simple and robust scenario for inducing magnetic vortex crystals in frustrated Mott insulators. By considering a quantum paramagnet that has a gapped spectrum with six-fold degenerate low-energy modes, we study the magnetic-field-induced condensation of these modes. We use a dilute gas approximation to demonstrate that a plethora of multi-Q condensates are stabilized for different combinations of exchange interactions. This rich quantum phase diagram includes magnetic vortex crystals, which are further stabilized by symmetric exchange anisotropies. Because skyrmion and domain-wall crystals have already been predicted and experimentally observed, this novel vortex phase completes the picture of emergent crystals of topologically nontrivial spin configurations.

Magnetization plateaus in quantum magnets—where bosonic quasiparticles crystallize into emergent spin superlattices—are spectacular yet simple examples of collective quantum phenomena escaping classical description. While magnetization plateaus have been observed in a number of spin-1/2 antiferromagnets, the description of their magnetic excitations remains an open theoretical and experimental challenge. Here, we investigate the dynamical properties of the triangular-lattice spin-1/2 antiferromagnet Ba3CoSb2O9 in its one-third magnetization plateau phase using a combination of nonlinear spin-wave theory and neutron scattering measurements. The agreement between our theoretical treatment and the experimental data demonstrates that magnons behave semiclassically in the plateau in spite of the purely quantum origin of the underlying magnetic structure. This allows for a quantitative determination of Ba3CoSb2O9 exchange parameters. We discuss the implication of our results to the deviations from semiclassical behavior observed in zero-field spin dynamics of the same material and conclude they must have an intrinsic origin.