Spin Ground State Research Articles

A Nanosized {NiII18} Cluster with a 'Flying Saucer' Topology Exhibiting Slow Relaxation of Magnetisation Phenomena at Both 15 K and 1.3 K.

A high-nuclearity {Ni18} complex (1) with a unique 'flying saucer' motif has been prepared from the organic chelate, α-methyl-2-pyridine-methanol (mpmH), in conjunction with bridging azido (N3-) and peroxido (O22-) ligands. Magnetic susceptibility measurements revealed the presence of both ferro- and antiferromagnetic exchange interactions between the metal centres in 1, and the stabilization of spin states with appreciable S values at two different temperature regimes. The end-on bridging azido and alkoxido groups are in all likelyhood the ferromagnetic mediators, while the η3:η3:μ6-bridging peroxides most likely promote the antiparallel alignment of the metals' spin vectors, yielding an overall non-zero spin ground state for the centrosymmetric compound 1. Furthermore, the {Ni18} nanosized cluster behaves as a single-molecule magnet, exhibiting magnetic hysteresis at low temperatures and two relaxation processes at 15 K and 1.3 K, a very rare phenomenon in polynuclear magnetic 3d-metal clusters.

Magnetoelectric Coupling in a Mn4Na-Organic Complex under Pulsed Magnetic Fields up to 73 T.

Research on the magnetoelectric (ME) effect (or spin-electric coupling) in molecule-based magnetic materials is a relatively nascent but promising topic. Molecule-based magnetic materials have diverse magnetic functionalities that can be coupled to electrical properties. Here we investigate a realization of ME coupling that is fundamental but not heavily studied─the coupling of magnetic spin level crossings to changes in electric polarization. A mixed-valence Mn4Na complex with a total ground-state spin S = 5/2 under zero magnetic field and S = 17/2 under high magnetic field undergoes a cascade of ground-state level crossings of the Sz states with increasing magnetic field. Magnetization and electrical polarization measurements under pulsed magnetic fields up to 73 T show that each spin level crossing is accompanied by a significant change in electric polarization that is an even function of the applied magnetic field. A molecular Hamiltonian describing antiferromagnetic exchange in a distorted tetrahedron of three MnIII and one MnII ions matches the data well. We conclude that the ME coupling is caused by magnetostriction within the polar molecule as it distorts to lower its magnetic exchange energy.

Block-Correlated Coupled Cluster Theory Based on the Generalized Valence Bond Reference for Singlet-Triplet Energy Gaps of Strongly Correlated Systems.

A block-correlated coupled cluster (BCCC) method based on the triplet generalized valence bond (GVB) wave function (GVB-BCCC) has been implemented for the first time. By introducing several techniques, we have developed a practical and efficient GVB-BCCC code. The GVB-BCCC3 method (with up to three-pair correlation) can be used to deal with strongly correlated (SC) systems with triplet or singlet ground states, allowing singlet-triplet (S-T) energy gaps in the active space of SC systems computationally available. For selected SC systems, our calculations show that GVB-BCCC3 can always provide correct ground-state spin multiplicity as the complete active space configuration interaction (CASCI) or density matrix renormalization group (DMRG). Furthermore, we found that the S-T energy gaps from GVB-BCCC3 are quite consistent with CASCI or DMRG results. This work demonstrates that GVB-BCCC3 is a promising theoretical tool for describing S-T energy gaps within the active space of SC systems with large active spaces.

Particle-Hole Asymmetric Ferromagnetism and Spin Textures in the Triangular Hubbard-Hofstadter Model

In a lattice model subject to a perpendicular magnetic field, when the lattice constant is comparable to the magnetic length, one enters the “Hofstadter regime,” where continuum Landau levels become fractal magnetic Bloch bands. Strong mixing between bands alters the nature of the resulting quantum phases compared to the continuum limit; lattice potential, magnetic field, and Coulomb interaction must be treated on equal footing. Using determinant quantum Monte Carlo and density matrix renormalization group techniques, we study this regime numerically in the context of the Hubbard-Hofstadter model on a triangular lattice. In the field-filling phase diagram, we find a broad wedge-shaped region of ferromagnetic ground states for filling factor ν≤1, bounded below by filling factor ν=1 and bounded above by half filling the lowest Hofstadter subband. We observe signatures of SU(2) quantum Hall ferromagnetism at filling factors ν=1 and ν=3. The phases near ν=1 are particle-hole asymmetric, and we observe a rapid decrease in ground-state spin polarization consistent with the formation of skyrmions only on the electron doped side. At large fields, above the ferromagnetic wedge, we observe a low-spin metallic region with spin correlations peaked at small momenta. We argue that the phenomenology of this region likely results from exchange interaction mixing fractal Hofstadter subbands. The phase diagram derived beyond the continuum limit points to a rich landscape to explore interaction effects in magnetic Bloch bands. Published by the American Physical Society 2024

Crystalline electric field excitations and their nonlinear splitting under magnetic fields in YbOCl

Recently reported van der Waals layered honeycomb rare-earth chalcohalides REChX (RE = rare earth, Ch = chalcogen, and X = halogen) are considered to be promising Kitaev spin liquid (KSL) candidates. The high-quality single crystals of YbOCl, a representative member of the family with an effective spin of 1/2, are available now. The crystalline electric field (CEF) excitations in a rare-earth spin system are fundamentally important for understanding both finite-temperature and ground-state magnetism but remain unexplored in YbOCl so far. In this paper, we conduct a comprehensive Raman scattering study to unambiguously identify the CEF excitations in YbOCl and determine the CEF parameters and wave functions. Our Raman experiments further reveal the anomalous nonlinear CEF splitting under magnetic fields. We have grown single crystals of YbOCl, the nonmagnetic LuOCl, and the diluted magnetic Lu0.86Yb0.14OCl to make a completely comparative investigation. Polarized Raman spectra on the samples at 1.8 K allow us to clearly assign all the Raman-active phonon modes and explicitly identify the CEF excitations in YbOCl. The CEF excitations are further examined using temperature-dependent Raman measurements and careful symmetry analysis based on Raman tensors related to CEF excitations. By applying the CEF Hamiltonian to the experimentally determined CEF excitations, we extract the CEF parameters and eventually determine the CEF wave functions. The study experimentally pins down the CEF excitations in the Kitaev compound YbOCl and sets a foundation for understanding its finite-temperature magnetism and exploring the possible nontrivial spin ground state. Published by the American Physical Society 2024

Regulating the Spin‐State of Cobalt in Three‐Dimensional Covalent Organic Frameworks for High‐Performance Sodium‐Iodine Rechargeable Batteries

Although the catalytic activity is heavily reliant on the electronic structure of the catalyst, understanding the impact of electron spin regulation on electrocatalytic performance is still rarely investigated. This work presents a novel approach involving the single‐atom coordination of cobalt (Co) within metalloporphyrin‐based three‐dimensional covalent organic frameworks (3D‐COFs) to facilitate the catalytic conversion for sodium‐iodine batteries. The spin state of Co is modulated by altering the oxidation state of the porphyrin‐centered Co, achieving optimal catalysis for iodine reduction. Experimental results demonstrate that CoII and CoIII are incorporated into the 3D‐COFs, exhibiting spin ground states of S = 1/2 and S = 0, respectively. The low spin state of CoIII is favorable to hybridize with the sp 3d orbitals of I3‐, thus facilitating the conversion of I3‐ to I‐. Density‐functional theory (DFT) calculations further reveal that the presence of CoIII enhances iodide adsorption and accelerates the formation of NaI in 3D‐COFs‐CoIII, thereby promoting its rapid kinetic behaviors. Notably, the I2@3D‐COFs‐CoIII cathode achieves a high reversible capacity of 227.7 mAh g‐1 after 200 cycles at 0.5 C and demonstrates exceptional cyclic stability, exceeding 2000 cycles at 10 C with a minor capacity fading rate of less than one 0.01% per cycle.

Regulating the Spin-State of Cobalt in Three-Dimensional Covalent Organic Frameworks for High-Performance Sodium-Iodine Rechargeable Batteries.

Although the catalytic activity is heavily reliant on the electronic structure of the catalyst, understanding the impact of electron spin regulation on electrocatalytic performance is still rarely investigated. This work presents a novel approach involving the single-atom coordination of cobalt (Co) within metalloporphyrin-based three-dimensional covalent organic frameworks (3D-COFs) to facilitate the catalytic conversion for sodium-iodine batteries. The spin state of Co is modulated by altering the oxidation state of the porphyrin-centered Co, achieving optimal catalysis for iodine reduction. Experimental results demonstrate that CoII and CoIII are incorporated into the 3D-COFs, exhibiting spin ground states of S = 1/2 and S = 0, respectively. The low spin state of CoIII is favorable to hybridize with the sp 3d orbitals of I3-, thus facilitating the conversion of I3- to I-. Density-functional theory (DFT) calculations further reveal that the presence of CoIII enhances iodide adsorption and accelerates the formation of NaI in 3D-COFs-CoIII, thereby promoting its rapid kinetic behaviors. Notably, the I2@3D-COFs-CoIII cathode achieves a high reversible capacity of 227.7 mAh g-1 after 200 cycles at 0.5 C and demonstrates exceptional cyclic stability, exceeding 2000 cycles at 10 C with a minor capacity fading rate of less than one 0.01% per cycle.

A [Mn8] Defective Supertetrahedron T3 and Its Dimeric [Mn16] Analogue.

The initial use of 2-(pyridin-2-yl)propane-1,3-diol (pypdH2) in Mn cluster chemistry has afforded two new mixed-valence polynuclear Mn clusters, namely, [Mn8Ο5(pypd)(hmp)3(O2CCMe3)8] (1) and [Mn16Ο10(N3)2(pypd)2{(py)2CO2}4(O2CEt)12] (2) (hmp- = deprotonated 2-hydroxymethylpyridine; (py)2CO2 2- = deprotonated gem-diol form of di-2-pyridyl ketone). Compound 1 features a novel [MnIII 7MnII(μ4-O)2(μ3-O)3(μ-OR)5]8+ structural core resembling a supertetrahedron T3, lacking two apexes, while complex 2 has a [MnIII 14MnII 2(μ4-O)4(μ3-O)6(μ-N3)2(μ3-OR)2(μ-OR)8]14+ core consisting of two [Mn8] subunits related to 1 and thus is a dimeric analogue of 1. Direct current (dc) magnetic susceptibility studies revealed the presence of dominant antiferromagentic exchange interactions between the Mn ions in complexes 1 and 2 and small ground state spin values for both compounds. Overall, this work highlights the capability of polyol-like chelates like pypdH2 to stabilize high nuclearity 3d metal clusters based on polynuclear building blocks.

Chemical Design of Spin Frustration to Realize Topological Spin Glasses.

Patterning spins to generate collective behavior is at the core of condensed matter physics. Physicists develop techniques, including the fabrication of magnetic nanostructures and precision layering of materials specifically to engender frustrated lattices. As chemists, we can access such exotic materials through targeted chemical synthesis and create new lattice types by chemical design. Here, we introduce a new approach to induce magnetic frustration on a modified honeycomb lattice through a competition of alternating antiferromagnetic (AFM) and ferromagnetic (FM) nearest-neighbor interactions. By subtly modulating these two types of interactions through facile synthetic modifications, we created two systems: (1) a topological spin glass and (2) a frustrated spin-canted magnet with low-temperature exchange bias. To design this unconventional magnetic lattice, we used a metal-organic framework (MOF) platform, Ni3(pymca)3X3 (NipymcaX where pymca = pyrimidine-2-carboxylato and X = Cl, Br). We isolated two MOFs, NipymcaCl and NipymcaBr, featuring canted Ni2+-based moments. Despite this similarity, differences in the single-ion anisotropies of the Ni2+ spins result in distinct magnetic properties for each material. NipymcaCl is a topological spin glass, while NipymcaBr is a rare frustrated magnet with low-temperature exchange bias. Density functional theory calculations and Monte Carlo simulations on the NipymcaX lattice support the presence of magnetic frustration as a result of alternating AFM and FM interactions. Our calculations enabled us to determine the ground-state spin configuration and the distribution of spin-spin correlations relative to paradigmatic kagomé and triangular lattices. This modified honeycomb lattice is similar to the electronic Kekulé-O phase in graphene and provides a highly tunable platform to realize unconventional spin physics.

Synthesis, general physicochemical behaviour and magnetic studies of the macrocyclic bis-pyridine derivatives [TTDPzM(py)2]·2H2O (M = MnII, CoII, NiII)

In our extensive work on phthalocyanine-like macrocycles carrying peripherally attached strongly electron-withdrawing 1,2,5-thia/selenodiazol rings we recently reported on the FeII species of formula [TTDPzFe(py)2].2H2O (TTDPz = tetrakis(thiadiazole)porphyrazinato dianion). The present work deals with the synthesis and extended physicochemical studies of the series of related macrocycles [TTDPzM(py)2].2H2O (M = MnII, CoII, NiII). The diffractograms of three solid samples of the NiII complex [TTDPzNi(py)2]·2H2O, obtained from different synthetic procedures, indicate that the species is reproducibly obtained. All three samples are isomorphous with each other as shown by their X-ray powder diffractograms. Based on the comparison of the IR spectra of [TTDPzNi(py)2].2H2O and the related desolvated [TTDPzNi], the IR absorption peaks belonging to the pyridine molecules could be identified. The systematic presence of the two water molecules in all three species suggests that forms of contact do exist in the form of hydrogen bonds between them and the S and N atoms present peripherally in the thiadiazole groups of the macrocyclic units. As regard to the elimination of water, the thermograms (in inert atmosphere in the range 25–600 °C) indicate in all cases that loss of water occurs below 100 °C but loss of pyridine occurs over a broad range 100–300 °C for the MnII and CoII species whereas for the NiII complex it takes place in the narrow range 200–220 °C. In relation to the parallel series of metallophthalocyanines, i.e. [PcM] (M = MnII, FeII, CoII), it is noted that the NiII analog has not been reported in the literature and in a direct experiment it has been proved that [PcNi] heated in pyridine is obtained unchanged, the different behavior for the NiII TTDPz derivative due to the electron withdrawing effect of the external thiadiazole substituents, which makes favorable axial pyridine coordination in the complex. Completely insoluble in water, the triad of TTDPz derivatives are extremely low-soluble in non-donor or low donor solvents. This precluded any attempt to isolate single crystals for single-crystal X-ray work. Their UV–visible spectra show essential similarities in whatever organic solvent is used with clearly positioned Soret (300–400 nm) and Q-band absorptions (630–650 nm). Variable temperature magnetic susceptibility and magnetization studies reveal that the [TTDPzM(py)2].2H2O complexes all show paramagnetism with ground state spins of 3/2 (MnII), ½ (CoII) and 1 (NiII), supported by fitting the data to spin Hamiltonian formalism, vide infra. Comparisons are made with the magnetism of [PcM] analogues with key differences noted for M = NiII.

{Fe3} to {Fe6} clusters through changing flexible 1,3-propanediol to rigid ethylene glycol: Syntheses, structures and magnetic properties

Two related FeIII complexes [Fe3(μ3-O)(O2CPh)6(H2L1)2(H2O)](NO3)·2C3H8O2 (1 {Fe3}) and [Fe6(μ3-O)2(O2CPh)6(HL2)6](NO3)2·2CH3CN (2 {Fe6}) (H2L1 = 1,3-propanediol and H2L2 = ethylene glycol) were synthesized and characterized. Complex 1 consists of a [Fe3O(PhCO2)3] unit. Changing ethylene glycol to 1,3-propanediol, the final cluster 2 comprises two [Fe3O(PhCO2)3] triangular units connected by six mono-deprotonated μ2−1η:2η ethylene glycol. The result indicates that the syntheses of FeIII complexes is sensitive to the ligands. Variable-temperature (T) and field (B) solid-state direct current magnetization (M) studies show that complexes 1 and 2 show strong anti-ferromagnetic coupling between FeIII ions, resulting in ground-state spins of S = 1/2 for 1 and S = 0 for 2, which is also confirmed by the calculations.

Luminescent Organic Triplet Diradicals as Optically Addressable Molecular Qubits.

Optical-spin interfaces that enable the photoinitialization, coherent microwave manipulation, and optical read-out of ground state spins have been studied extensively in solid-state defects such as diamond nitrogen vacancy (NV) centers and are promising for quantum information science applications. Molecular quantum bits (qubits) offer many advantages over solid-state spin centers through synthetic control of their optical and spin properties and their scalability into well-defined multiqubit arrays. In this work, we report an optical-spin interface in an organic molecular qubit consisting of two luminescent tris(2,4,6-trichlorophenyl)methyl (TTM) radicals connected via the meta-positions of a phenyl linker. The triplet ground state of this system can be photoinitialized in its |T0⟩ state by shelving triplet populations as singlets through spin-selective excited-state intersystem crossing with 80% selectivity from |T+⟩ and |T-⟩. The fluorescence intensity in the triplet manifold is determined by the ground-state polarization, and we show successful optical read-out of the ground-state spin following microwave manipulations by fluorescence-detected magnetic resonance spectroscopy. At 85 K, the lifetime of the polarized ground state is 45 ± 3 μs, and the ground state phase memory time is Tm = 5.9 ± 0.1 μs, which increases to 26.8 ± 1.6 μs at 5 K. These results show that luminescent diradicals with triplet ground states can serve as optically addressable molecular qubits with long spin coherence times, which marks an important step toward the rational design of spin-optical interfaces in organic materials.

Chemically Separable Co(II) Spin-State Isomers.

The phenomenon of spin crossover involves coordination complexes with switchable spin states. This spin state change is accompanied by significant geometric changes such that low and high spin forms of a complex are distinct isomers that exist in equilibrium with one another. Typically, spin-state isomers interconvert rapidly and are similar enough in polarity to prevent their independent separation and isolation. We report here the first example, to our knowledge, of cobalt(II) spin-state isomers that can be physically separated. The reaction of Mo2(dpa)4 (dpa = 2,2'-dipyridylamide) with CoBr2 produces a mixture of two heterometallic compounds with a linear, metal-metal-bonded Mo[Formula: see text]Mo-Co chain. The complexes, SC-[BrMo2(dpa)4Co]Br (SC-2) and HS-[BrMo2(dpa)4CoBr] (HS-2), have identical compositions (Mo2Co(dpa)4Br2) but different ground spin states and coordination geometries of the Co(II) ion. In the solid state, SC-2 undergoes incomplete spin crossover from an S = 1/2 state to an S = 3/2 state, and HS-2 has a high spin, S = 3/2, ground state, as confirmed by SQUID magnetometry and EPR spectroscopy. Crystallographic analyses of SC-2 and HS-2 show that SC-2 has an elongated Co-Br distance relative to HS-2 and is best described as the salt [BrMo2(dpa)4Co]Br. This limits SC-2's solubility in nonpolar solvents and allows for the physical separation of the two isomers. Solution studies of SC-2 and HS-2 indicate that SC-2 and HS-2 interconvert slowly relative to the NMR time scale. Additional solution-state EPR and UV-vis absorption measurements demonstrate that the choice of solvent polarity determines the predominant isomer present in solution.

Ground state magnetic structure and magnetic field effects in the layered honeycomb antiferromagnet YbOCl

YbOCl is a representative member of the van der Waals layered honeycomb rare-earth chalcohalide RChX (R = rare earth; Ch = O, S, Se, and Te; and X = F, Cl, Br, and I) family reported recently. Its spin ground state remains to be explored experimentally. We grew high-quality single crystals of YbOCl and conducted comprehensive thermodynamic, elastic, and inelastic neutron scattering experiments down to 50 mK. The experiments reveal an antiferromagnetic phase below 1.3 K which is identified as a spin ground state with an intralayer ferromagnetic and interlayer antiferromagnetic ordering. By applying sophisticated numerical techniques to a honeycomb (nearest-neighbor)–triangle (next-nearest-neighbor) model Hamiltonian which accurately describes the highly anisotropic spin system, we are able to simulate the experiments well and determine the diagonal and off-diagonal spin-exchange interactions. The simulations give an antiferromagnetic Kitaev term comparable to the Heisenberg one. The experiments under magnetic fields allow us to establish a magnetic field–temperature phase diagram around the spin ground state. Most interestingly, a relatively small magnetic field (∼0.3 to 3 T) can significantly suppress the antiferromagnetic order, suggesting an intriguing interplay of the Kitaev interaction and magnetic fields in the spin system. The present study provides fundamental insights into the highly anisotropic spin systems and opens a window to look into Kitaev spin physics in a rare-earth-based system. Published by the American Physical Society 2024

Tomonaga-Luttinger liquid and quantum criticality in spin- antiferromagnetic Heisenberg chain C 14 H 18 CuN 4 O 10 via Wilson ratio.

The ground state of a one-dimensional spin- uniform antiferromagnetic Heisenberg chain (AfHc) is a Tomonaga-Luttinger liquid which is quantum-critical with respect to applied magnetic fields up to a saturation field beyond which it transforms to a fully polarized state. Wilson ratio has been predicted to be a good indicator for demarcating these phases [Phys. Rev. B 96, 220401 (2017)]. From detailed temperature and magnetic field-dependent magnetization, magnetic susceptibility and specific heat measurements in a metalorganic complex and comparisons with field theory and quantum transfer matrix method calculations, the complex was found to be a very good realization of a spin- AfHc. Wilson ratio obtained from experimentally obtained magnetic susceptibility and magnetic contribution of specific heat values was used to map the magnetic phase diagram of the uniform spin- AfHc over large regions of phase space demarcating Tomonaga-Luttinger liquid, saturation field quantum critical, and fully polarized states. Luttinger parameter and spinon velocity were found to match very well with the values predicted from conformal field theory.

Theory-guided neural network for studying the ground state of 2D spin-orbit coupled Bose–Einstein condensates

Machine learning has been a powerful tool to study various models in many fields, which requires plenty of data to ensure accuracy. It is a challenge to reduce the dependence on the amount of data and ensure that the predictions are both accurate and physical. We develop a theory-guided neural network (TgNN) to explore the ground states of two-component spin–orbit coupled Bose–Einstein condensation in two-dimension. Only randomly selected few data from the twelve ground states as training data, TgNN performs higher accuracy, broader generalization ability and stronger robustness than deep neural network (DNN). Moreover, in the neighborhood of the phase transition where the predictions of DNN have obvious deviations, while TgNN still keeps relative accuracy. TgNN is less dependent on data and exhibits potential for high-dimensional and complicated models.

Alpha-decay half-lives and alpha-capture cross-sections

The same alpha-nucleus interaction describes the alpha-decay and alpha-capture reactions in the Unified Model for Alpha-Decay and Alpha-Capture (UMADAC). The data for the ground-state-to-ground-state alpha-transition half-lives in 420 nuclei, the half-lives for alpha-transition from the ground state to the first excited 2+ state in 74 even–even nuclei, and the alpha-capture cross sections of 20 spherical and deformed nuclei are used for defining the parameters of the UMADAC. The pointed data are well described in the framework of the UMADAC utilizing the obtained parameters. The ground-state-to-ground-state alpha-transition half-lives in 3802 nuclei with the proton Z and nucleon A numbers in the ranges 50 ≤Z≤ 126 and 97 ≤A≤ 340 are calculated in the UMADAC and presented in the Table 1. The quadrupole and hexadecapole deformations of the daughter nucleus are considered in the UMADAC. The minimal values of the orbital momenta of the alpha-transition between the ground states are found using available experimental and theoretical data for the ground state spin and parity values of nuclei. The half-lives for 223 alpha-transitions from the ground-state of the parent even–even nuclei to the lowest 2+ state of the daughter nuclei with the proton Z and nucleon A numbers in the ranges 52 ≤Z≤ 102 and 108 ≤A≤ 254 are obtained in the UMADAC and given in the Table 2.

A Monte Carlo study of Magnetic and thermodynamic behaviors of L[formula omitted] structure like quadrangular frustum pyramid nanoisland

Based on the Monte Carlo simulation, research has been conducted on the magnetic and thermodynamic behaviors of a ferrimagnetic mixed-spin (5/2, 1/2) L10 nanoisland with a like quadrangular frustum pyramid structure. We studied the effects of the magneto-crystal anisotropy and exchange coupling on the magnetization, susceptibility, internal energy, and hysteresis loops. The ferrimagnetic system exhibits more abundant magnetic and thermodynamic behavior than the cubic L10 nanoisland. We found eight saturation values of magnetization, five-loop hysteresis loops, and abrupt transitions in the low-temperature region. In addition, the exchange coupling effect of ferrimagnetism also promotes magnetic properties. Interestingly, in this system, the ferromagnetic exchange coupling (Jaa) and external magnetic field (h) linearly depend on the blocking temperature TB. The hysteresis loops of the system have more magnetization steps, which reflects that it has more ground state spin configurations.

Spin waves in antiferromagnetically coupled bilayers of transition-metal dichalcogenides with Dzyaloshinskii–Moriya interaction

In this paper we analyse quantized spin waves (also referred to as magnons) in bilayers of two-dimensional van der Waals materials, like Vanadium-based dichalcogenides, VX2 (X = S, Se, Te) and other materials of similar symmetry. We assume that the materials exhibit Dzyaloshinskii–Moriya interaction and in-plane easy-axis magnetic anisotropy due to symmetry breaking induced externally (e.g. by strain, gate voltage, proximity effects to an appropriate substrate/oberlayer, etc.). The considerations are limited to a collinear spin ground state, stabilized by a sufficiently strong in-plane magnetic anisotropy. The theoretical analysis is performed within the general spin wave theory based on the Holstein–Primakoff–Bogoliubov transformation. Accordingly, the description takes into account quantum antiferromagnetic fluctuations. However, it is limited to linear spinwave modes. The Dzyaloshinskii–Moriya interaction is shown to modify the spin wave spectrum of the bilayers, making its low energy part qualitatively similar to the electronic spectrum of the Rashba spin–orbit model.

Bessel vortices in spin-1 Bose–Einstein condensates with Zeeman splitting and spin–orbit coupling

Abstract We investigate the ground states of spin–orbit coupled spin-1 Bose–Einstein condensates in the presence of Zeeman splitting. By introducing the generalized momentum operator, the linear version of the system is solved exactly, yielding a set of Bessel vortices. Additionally, based on linear solution and using variational approximation, the solutions for the full nonlinear system and their ground state phase diagrams are derived, including the vortex states with quantum numbers m = 0, 1, as well as mixed states. In this work, mixed states in spin-1 spin–orbit coupling (SOC) BEC are interpreted for the first time as weighted superpositions of three vortex states. Furthermore, the results also indicate that under strong Zeeman splitting, the system cannot form localized states. The variational solutions align well with numerical simulations, showing stable evolution and meeting the criteria for long-term observation in experiments.