Zigzag Order Research Articles

Pressure-dependent magnetism of the Kitaev candidate Li2RhO3

We use magnetization measurements under pressure along with ab initio and cluster many-body calculations to investigate magnetism of the Kitaev candidate Li2RhO3. Hydrostatic compression leads to a decrease in the magnitude of the nearest-neighbor ferromagnetic Kitaev coupling K1 and the corresponding increase in the off-diagonal anisotropy Γ1, whereas the experimental Curie-Weiss temperature changes from negative to positive with the slope of +40 K/GPa. On the other hand, spin freezing persists up to at least 3.46 GPa with the almost constant freezing temperature of 5 K that does not follow the large changes in the exchange couplings and indicates the likely extrinsic origin of spin freezing. Magnetic frustration in Li2RhO3 is mainly related to the interplay between ferromagnetic K1 and antiferromagnetic Γ1, along with the weakness of the third-neighbor coupling J3 that would otherwise stabilize zigzag order. The small J3 distinguishes Li2RhO3 from other Kitaev candidates.

Anisotropy of the zigzag order in the Kitaev honeycomb magnet α−RuBr3

Kitaev materials often order magnetically at low temperatures due to the presence of non-Kitaev interactions. Torque magnetometry is a very sensitive technique for probing the magnetic anisotropy, which is critical in understanding the magnetic ground state. In this paper, we report detailed single-crystal torque measurements in the proposed Kitaev candidate honeycomb magnet α−RuBr3, which displays zigzag order below 34 K. Based on angular-dependent torque studies in magnetic fields up to 16 T rotated in the plane normal to the honeycomb layers, we find an easy-plane anisotropy with a temperature dependence of the torque amplitude following closely the behavior of the powder magnetic susceptibility. The torque for the field rotated in the honeycomb plane has a clear sixfold periodicity with a sawtooth shape, reflecting the threefold symmetry of the crystal structure and stabilization of different zigzag domains depending on the field orientation, with a torque amplitude that follows an order parameter form inside the zigzag phase. By comparing experimental data with theoretical calculations we highlight the importance of relevant anisotropic interactions and the role of the competition between different zigzag domains in this candidate Kitaev magnet. Published by the American Physical Society 2024

Low-energy spin dynamics in a Kitaev material Na3Ni2BiO6 investigated by nuclear magnetic resonance

Abstract We perform 23Na nuclear magnetic resonance (NMR) and magnetization measurements on an S = 1, quasi-2D honeycomb lattice antiferromagnet Na3Ni2BiO6. A large positive Curie–Weiss constant of 22.9 K is observed. The NMR spectra at low fields are consistent with a zigzag magnetic order, indicating a large easy-axis anisotropy. With the field applied along the c * axis, the NMR spectra confirm the existence of a 1/3-magnetization plateau phase between 5.1 T and 7.1 T. The transition from the zigzag order to the 1/3-magnetization plateau phase is also found to be a first-order type. A monotonic decrease of the spin gap is revealed in the 1/3-magnetization plateau phase, which reaches zero at a quantum critical field H C ≈ 8.35 T before entering the fully polarized phase. These data suggest the existence of exchange frustration in the system along with strong ferromagnetic interactions, hosting the possibility for Kitaev physics. Besides, well below the ordered phase, the 1/T 1 at high fields shows either a level off or an enhancement upon cooling below 3 K, which suggests the existence of low-energy fluctuations.

Mechanical characteristics and thermal conductivity of defect single-layer buckled honeycomb germanene

This study uses molecular dynamics (MD) simulation to investigate the defect rate, defect morphology, and different temperature effects on the mechanical properties, deformation behavior, and thermal conductivities of a single layer of germanene nanosheets via a tensile process. Samples are squeezed in the middle, leading to filling in minor defects. Young’s modulus and yield strength decrease with increasing temperature and defect rates. Young’s modulus in the armchair direction is larger than that in the zigzag direction, with the samples with a random porosity of 0%and 2% and smaller than the model with a random porosity of 4% to 10%. Young’s modulus in the armchair direction is larger than in the zigzag order with all the different pore shapes. The yield strength in the armchair direction is smaller than that in the zigzag at all temperatures, all different pore shapes, and all defect rates except for the sample with a random porosity of 2%. The thermal conductivity depends on the sample direction, the defect morphologies due to the shrinkage of membranes are complicated, and all are smaller than the thermal conductivity of a perfect sample. The thermal conductivity of the perfect sample is highest at 300 K.

Magnetic field-induced phase transition in ilmenite-type CoVO3

The ilmenite-type CoVO3 undergoes an antiferromagnetic transition below 140 K, and the tetravalent vanadium ions form V–V dimers below 550 K. This paper presents the magnetic spin structure and the phase transition induced by a magnetic field in CoVO3. Neutron powder diffraction at 50 K reveals that the spins of Co2+ arrange in a zigzag order on the honeycomb lattice with no external magnetic field. This magnetic spin structure does not exhibit spontaneous magnetization, corroborating previous magnetic measurements. Magnetization measurements in a pulsed-magnetic field revealed a transition to a weak-ferromagnetic phase at 37 T (100 K) and above 50 T (4.2 K), with spontaneous magnetization approximating 0.3–0.5 μB/f.u. This phase transition could potentially be attributed to a transition to a canted antiferromagnetic phase, which is brought about by the reduction in structural symmetry due to magnetostriction.

Regularity and Green’s relations for semigroups of transformations preserving a zigzag order and an equivalence

Consider a finite ordered set [Formula: see text] whose order forms a zigzag. For an equivalence [Formula: see text] on [Formula: see text], let [Formula: see text] be the set of all order homomorphism preserving [Formula: see text]. In this paper, we completely describe the regularity of elements and Green’s relations for [Formula: see text] under a certain condition.

Neutron Spectroscopy Evidence for a Possible Magnetic-Field-Induced Gapless Quantum-Spin-Liquid Phase in a Kitaev Material α-RuCl3

As one of the most promising Kitaev quantum-spin-liquid (QSL) candidates, α-RuCl3 has received a great deal of attention. However, its ground state exhibits a long-range zigzag magnetic order, which defies the QSL phase. Nevertheless, the magnetic order is fragile and can be completely suppressed by applying an external magnetic field. Here, we explore the evolution of magnetic excitations of α-RuCl3 under an in-plane magnetic field, by carrying out inelastic neutron scattering measurements on high-quality single crystals. Under zero field, there exist spin-wave excitations near the M point and a continuum near the Γ point, which are believed to be associated with the zigzag magnetic order and fractional excitations of the Kitaev QSL state, respectively. By increasing the magnetic field, the spin-wave excitations gradually give way to the continuous excitations. On the verge of the critical field μ 0 H c = 7.5 T, the former ones vanish and only the latter ones are left, indicating the emergence of a pure QSL state. By further increasing the field strength, the excitations near the Γ point become more intense. By following the gap evolution of the excitations near the Γ point, we are able to establish a phase diagram composed of three interesting phases, including a gapped zigzag order phase at low fields, possibly gapless QSL phase near μ 0 H c, and gapped partially polarized phase at high fields. These results demonstrate that an in-plane magnetic field can drive α-RuCl3 into a long-sought QSL state near the critical field.

Magnetic Properties of A2Ni2TeO6 (A = K, Li): Zigzag Order in the Honeycomb Layers of Ni2+ Ions Induced by First and Third Nearest-Neighbor Spin Exchanges.

The static and dynamic magnetic properties and the specific heat of K2Ni2TeO6 and Li2Ni2TeO6 were examined and it was found that they undergo a long-range ordering at TN = 22.8 and 24.4 K, respectively, but exhibit a strong short-range order. At high temperature, the magnetic susceptibilities of K2Ni2TeO6 and Li2Ni2TeO6 are described by a Curie–Weiss law, with Curie-Weiss temperatures Θ of approximately −13 and −20 K, respectively, leading to the effective magnetic moment of about 4.46 ± 0.01 μB per formula unit, as expected for Ni2+ (S = 1) ions. In the paramagnetic region, the ESR spectra of K2Ni2TeO6 and Li2Ni2TeO6 show a single Lorentzian-shaped line characterized by the isotropic effective g-factor, g = 2.19 ± 0.01. The energy-mapping analysis shows that the honeycomb layers of A2Ni2TeO6 (A = K, Li) and Li3Ni2SbO6 adopt a zigzag order, in which zigzag ferromagnetic chains are antiferromagnetically coupled, because the third nearest-neighbor spin exchanges are strongly antiferromagnetic while the first nearest-neighbor spin exchanges are strongly ferromagnetic, and that adjacent zigzag-ordered honeycomb layers prefer to be ferromagnetically coupled. The short-range order of the zigzag-ordered honeycomb lattices of K2Ni2TeO6 and Li2Ni2TeO6 is equivalent to that of an antiferromagnetic uniform chain, and is related to the short-range order of the ferromagnetic chains along the direction perpendicular to the chains.

Various regularities for semigroups of transformations on a finite set determined by a zig-zag order

Consider a finite ordered set [Formula: see text] with a zig-zag order. In this paper, we consider the semigroup [Formula: see text] of all compressing-preserving transformations on [Formula: see text]. Necessary and sufficient conditions for [Formula: see text] to be regular and coregular are given. Regular and coregular elements in [Formula: see text] are completely described. For a subsemigroup [Formula: see text] of the transformation semigroup [Formula: see text], we obtain that left regular and right regular elements in [Formula: see text] are identical. A characterization of left [right] regular elements in [Formula: see text] is given. We classify all left [right] regular semigroups [Formula: see text].

Magneto-optical study of metamagnetic transitions in the antiferromagnetic phase of \u03b1-RuCl3

α-RuCl3 is a promising candidate material to realize the so far elusive quantum spin liquid ground state. However, at low temperatures, the coexistence of different exchange interactions couple the effective pseudospins into an antiferromagnetically zigzag (ZZ) ordered state. The low-field evolution of spin structure is still a matter of debate and the magnetic anisotropy within the honeycomb planes is an open and challenging question. Here, we investigate the evolution of the ZZ order parameter by second-order magneto-optical effects, the magnetic linear dichroism and magnetic linear birefringence. Our results clarify the presence and nature of metamagnetic transitions in the ZZ phase of α-RuCl3. The experimental observations show the presence of initial magnetic domain repopulation followed by a spin-flop transition for small in-plane applied magnetic fields (≈1.6 T) along specific crystallographic directions. In addition, using a magneto-optical approach, we detected the recently reported emergence of a field-induced intermediate phase before suppressing the ZZ order. The results disclose the details of various angle-dependent in-plane metamagnetic transitions quantifying the bond-anisotropic interactions present in α-RuCl3.

Spin waves and Dirac magnons in a honeycomb-lattice zigzag antiferromagnet BaNi2(AsO4)2

The topological properties of massive and massless fermionic quasiparticles have been intensively investigated over the past decade in topological materials without magnetism. Recently, the bosonic analogs of such quasiparticles arising from spin waves have been reported in the two-dimensional (2D) honeycomb lattice ferromagnet/antiferromagnet and the 3D antiferromagnet. Here we use time-of-flight inelastic neutron scattering to study spin waves of the S = 1 honeycomb lattice antiferromagnet BaNi2(AsO4)2, which has a zig-zag antiferromagnetic (AF) ground state identical to that of the Kitaev quantum spin liquid candidate alpha-RuCl3. We determine the magnetic exchange interactions in the zig-zag AF ordered phase, and show that spin waves in BaNi2(AsO4)2 have symmetry-protected Dirac points inside the Brillouin zone boundary. These results provide a microscopic understanding of the zig-zag AF order and associated Dirac magnons in honeycomb lattice magnets, and are also important for establishing the magnetic interactions in Kitaev quantum spin liquid candidates.

Magnetoelastic coupling enabled tunability of magnon spin current generation in two-dimensional antiferromagnets

We theoretically investigate the magnetoelastic coupling (MEC) and its effect on magnon transport in two-dimensional antiferromagnets with a honeycomb lattice. MEC coeffcient along with magnetic exchange parameters and spring constants are computed for monolayers of transition metal trichalcogenides with N\'eel order ($\text{MnPS}_3$ and $\text{VPS}_3$) and zigzag order ($\text{CrSiTe}_3$, $\text{NiPS}_3$ and $\text{NiPSe}_3$) by $ab$ $initio$ calculations. Using these parameters, we predict that the spin-Nernst coefficient is significantly enhanced due to magnetoelastic coupling. Our study shows that although Dzyaloshinskii-Moriya interaction can produce spin Nernst effect in these materials, other mechanisms such as magnon-phonon coupling should be taken into account. We also demonstrate that the magnetic anisotropy is an important factor for control of magnon-phonon hybridization and enhancement of the Berry curvature and thus the spin-Nernst coefficient. Our results pave the way towards gate tunable spin current generation in 2D magnets by SNE via electric field modulation of MEC and anisotropy.

Machine-learned phase diagrams of generalized Kitaev honeycomb magnets

We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine (TK-SVM), to investigate the low-temperature classical phase diagram of a generalized Heisenberg-Kitaev-$\Gamma$ ($J$-$K$-$\Gamma$) model on a honeycomb lattice. Aside from reproducing phases reported by previous quantum and classical studies, our machine finds a hitherto missed nested zigzag-stripy order and establishes the robustness of a recently identified modulated $S_3 \times Z_3$ phase, which emerges through the competition between the Kitaev and $\Gamma$ spin liquids, against Heisenberg interactions. The results imply that, in the restricted parameter space spanned by the three primary exchange interactions -- $J$, $K$, and $\Gamma$, the representative Kitaev material $\alpha$-${\rm RuCl}_3$ lies close to the boundaries of several phases, including a simple ferromagnet, the unconventional $S_3 \times Z_3$ and nested zigzag-stripy magnets. A zigzag order is stabilized by a finite $\Gamma^{\prime}$ and/or $J_3$ term, whereas the four magnetic orders may compete in particular if $\Gamma^{\prime}$ is anti-ferromagnetic.

Observation of Giant Optical Linear Dichroism in a Zigzag Antiferromagnet FePS3.

Direct optical probing of the antiferromagnetic order parameter in atomically thin samples is challenging, for example, via magneto-optical spectroscopy, due to the lack of net magnetization. Here, we report zigzag-antiferromagnetism (AFM) induced optical linear dichroism (LD) in layered transition-metal thiophosphate FePS3 down to the monolayer limit. The observed LD is giant despite having the optical wave vector parallel to the Néel vector. The LD is at least one order of magnitude larger than those reported in other antiferromagnetic systems, where the optical wave vector is orthogonal to the Néel vector. The large LD enables the probe of 60° orientated zigzag-AFM domains. The optical anisotropy in FePS3 originates from an electronic anisotropy associated with the zigzag direction of the AFM order and is independent of the spin-pointing direction. Our findings point to a new optical approach for the investigation and control of zigzag or stripe magnetic order in strongly correlated systems.

Proximate ferromagnetic state in the Kitaev model material \u03b1-RuCl3

α-RuCl3 is a major candidate for the realization of the Kitaev quantum spin liquid, but its zigzag antiferromagnetic order at low temperatures indicates deviations from the Kitaev model. We have quantified the spin Hamiltonian of α-RuCl3 by a resonant inelastic x-ray scattering study at the Ru L3 absorption edge. In the paramagnetic state, the quasi-elastic intensity of magnetic excitations has a broad maximum around the zone center without any local maxima at the zigzag magnetic Bragg wavevectors. This finding implies that the zigzag order is fragile and readily destabilized by competing ferromagnetic correlations. The classical ground state of the experimentally determined Hamiltonian is actually ferromagnetic. The zigzag state is stabilized by quantum fluctuations, leaving ferromagnetism – along with the Kitaev spin liquid – as energetically proximate metastable states. The three closely competing states and their collective excitations hold the key to the theoretical understanding of the unusual properties of α-RuCl3 in magnetic fields.

Identification of magnetic interactions and high-field quantum spin liquid in \u03b1-RuCl3

The frustrated magnet α-RuCl3 constitutes a fascinating quantum material platform that harbors the intriguing Kitaev physics. However, a consensus on its intricate spin interactions and field-induced quantum phases has not been reached yet. Here we exploit multiple state-of-the-art many-body methods and determine the microscopic spin model that quantitatively explains major observations in α-RuCl3, including the zigzag order, double-peak specific heat, magnetic anisotropy, and the characteristic M-star dynamical spin structure, etc. According to our model simulations, the in-plane field drives the system into the polarized phase at about 7 T and a thermal fractionalization occurs at finite temperature, reconciling observations in different experiments. Under out-of-plane fields, the zigzag order is suppressed at 35 T, above which, and below a polarization field of 100 T level, there emerges a field-induced quantum spin liquid. The fractional entropy and algebraic low-temperature specific heat unveil the nature of a gapless spin liquid, which can be explored in high-field measurements on α-RuCl3.

Oscillations of the thermal conductivity in the spin-liquid state of α-RuCl3

In the class of materials called spin liquids, a magnetically ordered state cannot be attained even at milliKelvin temperatures because of conflicting constraints on each spin (for e.g. from geometric or exchange frustration). The resulting quantum spin-liquid (QSL) state is currently of intense interest because it exhibits novel excitations as well as wave-function entanglement. The layered insulator $\alpha$-RuCl$_3$ orders as a zigzag antiferromagnet below $\sim$7 K in zero magnetic field. The zigzag order is destroyed when a magnetic field $\bf H$ is applied parallel to the zigzag axis a. Within the field interval (7.3, 11) Tesla, there is growing evidence that a QSL state exists. Here we report the observation of oscillations in its thermal conductivity below 4 K. The oscillation amplitude is very large within the interval (7.3, 11) T and strongly suppressed on either side. Paradoxically, the oscillations are periodic in 1/\emph{H}, analogous to quantum oscillations in metals, even though $\alpha$-RuCl$_3$ is an excellent insulator with a gap of 1.9 eV. By tilting $\bf H$ out of the plane, we find that the oscillation period is determined by the in-plane component $H_a$. As the temperature is raised above 0.5 K, the oscillation amplitude decreases exponentially. The decrease anticorrelates with the emergence above $\sim$2 K of an anomalous planar thermal Hall conductivity measured with $\bf H\parallel a$. To exclude extrinsic artifacts, we carried out several tests. The implications of the oscillations are discussed.

High‐quality reversible data hiding scheme using sorting and enhanced pairwise PEE

This paper proposes a novel reversible data hiding (RDH) technique using sorting and pairwise prediction error expansion (PEE) to improve embedding capacity (EC) while retaining the quality of cover image. The proposed scheme traverses alternate pixels of the cover image in a zig-zag order to construct two independent sets for sequential embedding. The pixels of each set are sorted in an increasing order of their rhombus mean followed by a two-pass data embedding by dividing the sets into 1 × 3 size blocks based on some pre-defined criteria. In pass 1, two prediction errors are calculated for the first and the last pixels using their rhombus means; and pairwise mapping is modified and exploited to embed the secret data in such a way that the value of the first pixel is either increased or remains unchanged, and the value of the last pixel is either decreased or remains unchanged. In pass-2, the middle pixel is utilised to predict the first and last pixels and the values of the first pixel and the last pixel are either decreased or remains unchanged and either increased or remains unchanged, respectively. In contrast to some existing recovery-based methods, the proposed pass-2 guarantees to complement the changes made in pass-1, thereby boosting the quality along with increased EC. The experimental results show that the proposed method achieves better embedding performance than the state-of-the-art RDH techniques. More specifically, the proposed method gets an increment by an average of 0.46 dB and 1.01 dB for embedding 10,000 bits and 20,000 bits respectively over its closest prior art.

Symmetry enhanced spin-Nernst effect in honeycomb antiferromagnetic transition metal trichalcogenide monolayers

We investigate systematically the spin-Nernst effect in N\'eel and zigzag ordered honeycomb antiferromagnets. Monolayers of transition-metal trichalcogenides, $\mathrm{Mn}\mathrm{P}{\mathrm{Se}}_{3}$, $\mathrm{Mn}{\mathrm{PS}}_{3}$, and ${\text{VPS}}_{3}$ show an antiferromagnetic N\'eel order while $\mathrm{Cr}\mathrm{Si}{\mathrm{Te}}_{3}$, $\mathrm{Ni}{\mathrm{PS}}_{3}$, and $\mathrm{Ni}\mathrm{P}{\mathrm{Se}}_{3}$ show an antiferromagnetic zigzag order. We extract the exchange and Dzyaloshinskii-Moriya interaction parameters from ab initio calculations. Using these parameters, we predict that the spin-Nernst coefficient is at least two orders of magnitude larger in zigzag compared to the N\'eel ordered antiferromagnets. We find that this enhancement relies on the large band splitting due to the symmetry of magnetic configuration in the zigzag order. Our calculations indicate that the Dzyaloshinskii-Moriya interaction is the underlying factor for the spin-Nernst effect in both cases, although with different microscopic mechanisms. In the case of N\'eel antiferromagnets, magnon bands already possess a Berry curvature and introducing the Dzyaloshinskii-Moriya interaction splits the magnon bands with the opposite helicity throughout the Brillouin zone which results in an unbalanced population of magnons carrying opposite spins. In the case of zigzag antiferromagnets, magnon bands do not possess the Berry curvature but they are split for opposite helicity magnons due to symmetry of the system. In this case, introducing the Dzyaloshinskii-Moriya interaction induces the Berry curvature and results in the spin-Nernst effect. Due to large magnon band splitting, the spin-Nernst effect in zigzag antiferromagnets is stronger than N\'eel antiferromagnets.

Generating sets of an infinite semigroup of transformations preserving a zig-zag order

A zig-zag order is like a directed path, only with alternating directions. A generating set of minimal size for the semigroup of all full transformations on a finite set preserving the zig-zag order was determined by Fenandes et al. in 2019. This paper deals with generating sets of the semigroup $F_{\mathbb{N}}$ of all full transformations on the set of all natural numbers preserving the zig-zag order. We prove that $F_{\mathbb{N}}$ has no minimal generating sets and present two particular infinite decreasing chains of generating sets of $F_{\mathbb{N}}.$