Polarization-Addressable Optical Movement of Plasmonic Nanoparticles and Hotspot Spin Vortices

Spin-orbit coupling in nanoscale optical fields leads to the emergence of a nontrivial spin angular momentum component, transverse to the orbital momentum. In this study, we initially investigate how this spin-orbit coupling effect influences the dynamics in gold monomers. We observe that localized surface plasmon resonance induces self-generated transverse spin, affecting the trajectory of the nanoparticles as a function of the incident polarization. Furthermore, we investigate the spin-orbit coupling in gold dimers. The resonant spin momentum distribution is characterized by the unique formation of vortex and anti-vortex spin angular momentum pairs on opposite surfaces of the nanoparticles, also affecting the particle motion. These findings hold promise for various fields, particularly for the precision control in the development of plasmonic thrusters and the development of metasurfaces and other helicity-controlled system aspects. They offer a method for the development of novel systems and applications in the realm of spin optics.

Spin-orbit torques due to topological insulator surface states: an in-plane magnetization as a probe of extrinsic spin-orbit scattering

Topological insulator (TI) surface states exert strong spin-orbit torques. When the magnetization is in the plane its interaction with the TI conduction electrons is non-trivial, and is influenced by extrinsic spin-orbit scattering. This is expected to be strong in TIs but is difficult to calculate and to measure unambiguously. Here we show that extrinsic spin-orbit scattering sizably renormalizes the surface state spin-orbit torque resulting in a strong density dependence. The magnitude of the renormalization of the spin torque and the effect of spin-orbit scattering on the relative sizes of the in-plane and out-of-plane field-like torques have strong implications for experiment: We propose two separate experimental signatures for the measurement of its presence.

Probing the Magnetic Ground State of Ba2YIrO6: Impact of Nonmagnetic Dopants and Spin-Orbit Coupling

Strong spin-orbit coupling (SOC) in iridates has long been predicted to lead to exotic electronic and magnetic ground states. Ba2YIrO6 (BYIO) has attracted particular attention due to the expectation of a Jeff = 0 state for Ir5+ ions under the jj-coupling scheme. However, controversies surround the actual realization of this state, as finite magnetic moments are consistently observed experimentally. We present a multi-physics study of this system by progressively introducing nonmagnetic Sb5+ ions in place of Ir5+ (Ba2YIr​1-ySb​yO​6, BYISO). Despite similar charge and ionic radii, Sb​5+ doping appears highly inhomogeneous, coexisting with a fraction of nearly pure BYIO regions, as confirmed by X-ray diffraction (XRD). This aligns with observations in related compounds. While inhomogeneity creates uncertainty, the doped majority phases offer valuable insights. It is relevant that the inclusion of even small amounts of Sb​5+ (10-20%) leads to a rise in magnetization. This strengthens our previous suggestion that magnetic Ir ions form dynamic singlets in BYIO, resulting in a near-nonmagnetic background. The observed moment enhancement with nonmagnetic doping supports the breakdown of these singlets. Furthermore, the magnetization steadily increases with an increasing Sb​5+ content, contradicting the anticipated approach towards the Jeff = 0 state with increased SOC due to reduced hopping between Ir​5+ ions. This reinforces the presence of individual Ir​5+ moments. Overall, our findings suggest that Ba​2YIrO​6 might not possess sufficiently strong SOC to be solely described within the jj-coupling picture, paving the way for further investigation.

Antiferromagnetically ordered topological semimetals in Hubbard model with spin-orbit coupling

We examine the possible existence of Dirac semimetal with magnetic order in a two-dimensional system with a nonsymmorphic symmetry by using the Hartree-Fock mean-field theory within the Hubbard model. We locate the region in the second-neighbor spin-orbit coupling vs Hubbard interaction phase diagram, where such a state is stabilized. The edge states for the ribbons along two orthogonal directions concerning the orientation of in-plane magnetic moments are obtained. Finally, the effect of the in-plane magnetic field, which results in the stabilization of the Weyl semimetallic (WSM) state, and the nature of the edge states corresponding to the WSM state for ribbon geometries are also explored.

Atomic-Layer Controlled Transition from Inverse Rashba-Edelstein Effect to Inverse Spin Hall Effect in 2D PtSe2 Probed by THz Spintronic Emission

2D materials, such as transition metal dichalcogenides, are ideal platforms for spin-to-charge conversion (SCC) as they possess strong spin-orbit coupling (SOC), reduced dimensionality and crystal symmetries as well as tuneable band structure, compared to metallic structures. Moreover, SCC can be tuned with the number of layers, electric field, or strain. Here, SCC in epitaxially grown 2D PtSe2 by THz spintronic emission is studied since its 1T crystal symmetry and strong SOC favor SCC. High quality of as-grown PtSe2 layers is demonstrated, followed by in situ ferromagnet deposition by sputtering that leaves the PtSe2 unaffected, resulting in well-defined clean interfaces as evidenced with extensive characterization. Through this atomic growth control and using THz spintronic emission, the unique thickness-dependent electronic structure of PtSe2 allows the control of SCC. Indeed, the transition from the inverse Rashba-Edelstein effect (IREE) in 1-3 monolayers (ML) to the inverse spin Hall effect (ISHE) in multilayers (>3 ML) of PtSe2 enabling the extraction of the perpendicular spin diffusion length and relative strength of IREE and ISHE is demonstrated. This band structure flexibility makes PtSe2 an ideal candidate to explore the underlying mechanisms and engineering of the SCC as well as for the development of tuneable THz spintronic emitters.

The Spin-Orbit Effect on the Electronic Structures, Refractive Indices, and Birefringence of X2PO4I (X = Pb, Sn, Ba and Sr): A First-Principles Investigation

Introducing post-transition metal cations is an excellent strategy for enhancing optical properties. This paper focuses on four isomers, namely the X2PO4I (X = Pb, Sn, Ba, and Sr) series. For the first time, the paper's attention is paid to the changes in electronic structure, as well as refractive indices and birefringence, with and without the inclusion of spin-orbit effects in this series. The first-principles results show that spin-orbit effects of the 5p and 6p states found in these compounds lead to splitting of the bands, narrowing of the band gap, enhancement of the lone-pair stereochemistry, and enhancement of the refractive indices and birefringence. Moreover, a comparison of the lone-pair electron phosphates, X2PO4I (X = Pb and Sn), and the isomeric alkaline earth metal phosphates, X2PO4I (X = Ba and Sr), reveals that changes in the band structure have a greater effect on the enhancement of the birefringence than the slight enhancement of the lone-pair stereochemical activity. This study has important implications for a deeper understanding of the optical properties of crystals and the design of novel optical materials.

Spin-photon interaction in a nanowire quantum dot with asymmetrical confining potential

The electron (hole) spin-photon interaction is studied in an asymmetrical InSb (Ge) nanowire quantum dot. The spin-orbit coupling in the quantum dot mediates not only a transverse spin-photon interaction, but also a longitudinal spin-photon interaction due to the asymmetry of the confining potential. Both the transverse and the longitudinal spin-photon interactions have non-monotonic dependence on the spin-orbit coupling. For realistic spin-orbit coupling in the quantum dot, the longitudinal spin-photon interaction is much (at least one order) smaller than the transverse spin-photon interaction. The order of the transverse spin-photon interaction is about 1 nm in terms of length|zeg|, or 0.1 MHz in terms of frequencyeE0|zeg|/hfor a moderate cavity electric field strengthE0=0.4V m-1.

Discrete and Semi-Discrete Multidimensional Solitons and Vortices: Established Results and Novel Findings

This article presents a concise survey of basic discrete and semi-discrete nonlinear models, which produce two- and three-dimensional (2D and 3D) solitons, and a summary of the main theoretical and experimental results obtained for such solitons. The models are based on the discrete nonlinear Schrödinger (DNLS) equations and their generalizations, such as a system of discrete Gross-Pitaevskii (GP) equations with the Lee-Huang-Yang corrections, the 2D Salerno model (SM), DNLS equations with long-range dipole-dipole and quadrupole-quadrupole interactions, a system of coupled discrete equations for the second-harmonic generation with the quadratic (χ(2)) nonlinearity, a 2D DNLS equation with a superlattice modulation opening mini-gaps, a discretized NLS equation with rotation, a DNLS coupler and its PT-symmetric version, a system of DNLS equations for the spin-orbit-coupled (SOC) binary Bose-Einstein condensate, and others. The article presents a review of the basic species of multidimensional discrete modes, including fundamental (zero-vorticity) and vortex solitons, their bound states, gap solitons populating mini-gaps, symmetric and asymmetric solitons in the conservative and PT-symmetric couplers, cuspons in the 2D SM, discrete SOC solitons of the semi-vortex and mixed-mode types, 3D discrete skyrmions, and some others.

Beyond Kitaev physics in strong spin-orbit coupled magnets

We review the recent advances and current challenges in the field of strong spin-orbit coupled Kitaev materials, with a particular emphasis on the physics beyond the exactly-solvable Kitaev spin liquid point. To this end, we present a comprehensive overview of the key exchange interactions in candidate materials with a specific focus on systems featuring effectiveJeff=1/2magnetic moments. This includes, but not limited to,5d5iridates,4d5ruthenates and3d7cobaltates. Our exploration covers the microscopic origins of these interactions, along with a systematic attempt to map out the most intriguing correlated regimes of the multi-dimensional parameter space. Our approach is guided by robust symmetry and duality transformations as well as insights from a wide spectrum of analytical and numerical studies. We also survey higher spin Kitaev models and recent exciting results on quasi-one-dimensional models and discuss their relevance to higher-dimensional models. Finally, we highlight some of the key questions in the field as well as future directions.

Evaluation of Sputtering Processes in Strontium Iridate Thin Films

The growth of epitaxial thin films from the Ruddlesden-Popper series of strontium iridates by magnetron sputtering is analyzed. It was found that, even using a non-stoichiometric target, the films formed under various conditions were consistently of the perovskite-like n = ∞ SrIrO3 phase, with no evidence of other RP series phases. A detailed inspection of the temperature-oxygen phase diagram underscored that kinetics mechanisms prevail over thermodynamics considerations. The analysis of the angular distribution of sputtered iridium and strontium species indicated clearly different spatial distribution patterns. Additionally, significant backsputtering was detected at elevated temperatures. Thus, it is assumed that the interplay between these two kinetic phenomena is at the origin of the preferential nucleation of the SrIrO3 phase. In addition, strategies for controlling cation stoichiometry off-axis have also been explored. Finally, the long-term stability of the films has been demonstrated.

Defected twisted ring topology for autonomous periodic flip-spin-orbit soft robot

Periodic spin-orbit motion is ubiquitous in nature, observed from electrons orbiting nuclei to spinning planets orbiting the Sun. Achieving autonomous periodic orbiting motions, along circular and noncircular paths, in soft mobile robotics is crucial for adaptive and intelligent exploration of unknown environments-a grand challenge yet to be accomplished. Here, we report leveraging a closed-loop twisted ring topology with a defect for an autonomous soft robot capable of achieving periodic spin-orbiting motions with programmed circular and re-programmed irregular-shaped trajectories. Constructed by bonding a twisted liquid crystal elastomer ribbon into a closed-loop ring topology, the robot exhibits three coupled periodic self-motions in response to constant temperature or constant light sources: inside-out flipping, self-spinning around the ring center, and self-orbiting around a point outside the ring. The coupled spinning and orbiting motions share the same direction and period. The spinning or orbiting direction depends on the twisting chirality, while the orbital radius and period are determined by the twisted ring geometry and thermal actuation. The flip-spin and orbiting motions arise from the twisted ring topology and a bonding site defect that breaks the force symmetry, respectively. By utilizing the twisting-encoded autonomous flip-spin-orbit motions, we showcase the robot's potential for intelligently mapping the geometric boundaries of unknown confined spaces, including convex shapes like circles, squares, triangles, and pentagons and concaves shapes with multi-robots, as well as health monitoring of unknown confined spaces with boundary damages.

Electronic Structures of Kitaev Magnet Candidates RuCl3 and RuI3

Layered honeycomb magnets with strong atomic spin-orbit coupling at transition metal sites have been intensively studied for the search of Kitaev magnetism and the resulting non-Abelian braiding statistics. α-RuCl3 has been the most promising candidate, and there have been several reports on the realization of sibling compounds α-RuBr3 and α-RuI3 with the same crystal structure. Here, we investigate correlated electronic structures of α-RuCl3 and α-RuI3 by employing first-principles dynamical mean-field theory. Our result provides a valuable insight into the discrepancy between experimental and theoretical reports on transport properties of α-RuI3, and suggests a potential realization of correlated flat bands with strong spin-orbit coupling and a quantum spin-Hall insulating phase in α-RuI3.

The Einstein-de Haas effect in an Fe15cluster

Classical models of spin-lattice coupling are at present unable to accurately reproduce results for numerous properties of ferromagnetic materials, such as heat transport coefficients or the sudden collapse of the magnetic moment in hcp-Fe under pressure. This inability has been attributed to the absence of a proper treatment of effects that are inherently quantum mechanical in nature, notably spin-orbit coupling (SOC). This paper introduces a time-dependent, non-collinear tight binding model, complete with SOC and vector Stoner exchange terms, that is capable of simulating the Einstein-de Haas (EdH) effect in a ferromagnetic Fe15cluster. The tight binding model is used to investigate the adiabaticity timescales that determine the response of the orbital and spin angular momenta to a rotating, externally appliedBfield, and we show that the qualitative behaviors of our simulations can be extrapolated to realistic timescales by use of the adiabatic theorem. An analysis of the trends in the torque contributions with respect to the field strength demonstrates that SOC is necessary to observe a transfer of angular momentum from the electrons to the nuclei at experimentally realisticBfields. The simulations presented in this paper demonstrate the EdH effect from first principles using a Fe cluster.

Compass-like manipulation of electronic nematicity in Sr3Ru2O7

Electronic nematicity has been found in a wide range of strongly correlated electron materials, resulting in the electronic states having-4.5pc]Please note that the spelling of the following author name(s) in the manuscript differs from the spelling provided in the article metadata: Izidor Benedičič. The spelling provided in the manuscript has been retained; please confirm. a symmetry that is lower than that of the crystal that hosts them. One of the most astonishing examples is [Formula: see text], in which a small in-plane component of a magnetic field induces significant resistivity anisotropy. The direction of this anisotropy follows the direction of the in-plane field. The microscopic origin of this field-induced nematicity has been a long-standing puzzle, with recent experiments suggesting a field-induced spin density wave driving the anisotropy. Here, we report spectroscopic imaging of a field-controlled anisotropy of the electronic structure at the surface of [Formula: see text]. We track the electronic structure as a function of the direction of the field, revealing a continuous change with the angle. This continuous evolution suggests a mechanism based on spin-orbit coupling resulting in compass-like control of the electronic bands. The anisotropy of the electronic structure persists to temperatures about an order of magnitude higher compared to the bulk, demonstrating novel routes to stabilize such phases over a wider temperature range.

Coherent magnetic and electronic structure symmetry broken in frustrated bilayer Kagome ferromagnet Fe3Sn2

In this letter, by measuring resistivity and magnetization with magnetic fieldHrotated inabplane and currentIalongcaxis, we studied the magnetic and electronic structure symmetry of frustrated topological bilayer Kagome ferromagnet Fe3Sn2. We observed that the curves of the resistivity and magnetization both showed broken two-fold symmetry from 5 K to 380 K. The further analysis indicates that there is a close causality between the spin arrangement and the electronic states in Fe3Sn2even above room temperature. These phenomena are closely related to the change in spin-orbit coupling (SOC) under the magnetic field. Our experimental results suggest that Fe3Sn2is an ideal platform to study the influence of spin arrangement on electronic state in topological materials and can also be used to design next generation magnetic devices by modulating SOC at external magnetic field.

Optical properties of methyl-substituted germanane monolayer in the presence of the external magnetic field, strain and spin-orbit coupling

In this research, we use the tight-binding model, which includes spin-orbit coupling and an external magnetic field, to describe the optical properties of the methyl-substituted germanane (GeCH3) monolayer. We have applied the Kubo formula, linear response theory, and Green's function approach to calculate the optical absorption coefficient of the GeCH3monolayer. Here, the effects of an external magnetic field, strain, spin-orbit coupling, temperature, and electron/hole doping on the frequency dependence behavior of the optical conductivity have been investigated in detail. Our numerical results show that with increasing the external magnetic field, strain, and electron doping, the weight of the Drude increases. The optical absorption peak decreases and shifts to higher frequencies by applying an external magnetic field, strain, and electron doping. Controlling the optical and electronic properties of GeCH3is leading to use this structure it in the electronic and optoelectronic industries.

Existence of nodal-arc and its evolution into Weyl-nodes in the presence of spin-orbit coupling in TaAs & TaP

In this work, we report the existence ofnodal-arc, which acts as the building block of all the nodal-rings in TaAs & TaP. Thisnodal-arcis found to be capable of generating all the nodal-rings in these materials upon the application of space-group symmetry operations including time-reversal symmetry. The arcs are obtained to be dispersive with the energy spread of ∼109 (∼204) meV in TaAs (TaP). Also, the orbitals leading to bands-inversion and thus the formation ofnodal-arcsare found to be Ta-5d& As-4p(P-3p) in TaAs (TaP). The area of nodal-rings is found to be highly sensitive to the change in hybridization-strength, where the increase in hybridization-strength leads to the decrease in the area of nodal-rings. In the presence of spin-orbit coupling (SOC), all the points on these arcs get gaped-up and two pairs of Weyl-nodes are found to evolve from them. Out of the two pair, one is found to be situated close to the joining point of the two arcs forming a ring. This causes the evolution of each nodal-ring into three pairs of Weyl-nodes. The coordinates of these Weyl-nodes are found to be robust to the increase in SOC-strength from ∼0.7-3.5 eV. All the results are obtained at thefirst-principlelevel. This work provides a clear picture of the existence ofnodal-arcdue to accidental degeneracy and its evolution into Weyl-nodes under the effect of SOC.

First-principles predictions of enhanced thermoelectric properties for Cs2SnI2Cl2and Cs2PbI2Cl2monolayers with spin-orbit coupling

Inspired by the exceptional charge transport properties and ultra-low thermal conductivity of halide perovskite, we investigate the electronic nature, thermal transport, and thermoelectric properties for Ruddlesden-Popper all-inorganic perovskite, Cs2SnI2Cl2and Cs2PbI2Cl2monolayers, using first-principles calculations. During the calculations, spin-orbit coupling has been considered for electronic transport as well as thermoelectric properties. The results show that the Cs2SnI2Cl2and Cs2PbI2Cl2monolayers exhibit high carrier mobility and low thermal conductivity. Stronger phonon-phonon interaction is responsible for the fact that thermal conductivity of Cs2SnI2Cl2monolayer is much lower than that of Cs2PbI2Cl2monolayer. At 700 K, the values of the figure of merit (ZT) for the n-type doped Cs2SnI2Cl2and Cs2PbI2Cl2monolayers are about 1.05 and 0.32 at the optimized carrier concentrations 5.42 × 1012cm-2and 9.84 × 1012cm-2. Moreover, when spin-orbit coupling is considered, the correspondingZTvalues are enhanced to 2.73 and 1.98 at 5.27 × 1011cm-2and 6.16 × 1011cm-2. These results signify that Cs2SnI2Cl2and Cs2PbI2Cl2monolayers are promising thermoelectric candidates.

Two ground-state bright solitons in fractional-order spin-orbit-coupled Bose-Einstein condensates

We study two types of bright solitons in zero and non-zero detuning spin-orbit-coupled (SOC) Bose-Einstein condensates in fractional effect by variational and imaginary-time evolution method. The results show that the variation of SOC with fractional kinetic energy operator affects the existence and form of solitons. In particular, we found that SOC strengthγ=1.0could be a transition point. In other words, forγ<1.0andγ⩾1.0, the soliton states and pseudo-spin polarization in non-zero detuning show opposite changes. In addition, moving bright solitons with zero and non-zero detuning are studied.

Electron-phonon coupling in La3In under pressure

Conventional superconducting parameters Eliashberg spectral function, phonon DOS, average electron-phonon coupling constant & superconducting transition temperature of La₃In were calculated at ambient and applied positive hydrostatic pressure. Ambient pressure electron-phonon coupling constant was found to be 1.43 and superconducting transition temperature was calculated to be 8.7 K which is close to the experimental reported value of 9.5 K. Logarithmic average phonon frequency at ambient pressure was found to be 6.21 × 10^-3eV. Phonon dispersion relations for the ambient and applied positive hydrostatic pressure were calculated to confirm the dynamical stability of La₃In under positive pressure. Acoustic modes of vibrations were found to mediate most of the electron-phonon-electron interaction with a very small contribution from optical modes of vibrations. Fermi surface at ambient pressure was also calculated and employed in explaining the large value of the calculated electron-phonon coupling constant. The effect of spin-orbit coupling on electronic, phonon and superconducting properties of La₃In was also studied.

Chiral photocurrent in a Quasi-1D TiS3(001) phototransistor

The presence of in-plane chiral effects, hence spin-orbit coupling, is evident in the changes in the photocurrent produced in a TiS₃(001) field-effect phototransistor with left versus right circularly polarized light. The direction of the photocurrent is protected by the presence of strong spin-orbit coupling and the anisotropy of the band structure as indicated in NanoARPES measurements. Dark electronic transport measurements indicate that TiS₃is n-type and has an electron mobility in the range of 1-6 cm²V^-1s^-1.I-Vmeasurements under laser illumination indicate the photocurrent exhibits a bias directionality dependence, reminiscent of bipolar spin diode behavior. Because the TiS₃contains no heavy elements, the presence of spin-orbit coupling must be attributed to the observed loss of inversion symmetry at the TiS₃(001) surface.

Impact of Bismuth Incorporation into (Ga,Mn)As Dilute Ferromagnetic Semiconductor on Its Magnetic Properties and Magnetoresistance

The impact of bismuth incorporation into the epitaxial layer of a (Ga,Mn)As dilute ferromagnetic semiconductor on its magnetic and electromagnetic properties is studied in very thin layers of quaternary (Ga,Mn)(Bi,As) compound grown on a GaAs substrate under a compressive misfit strain. An addition of a small atomic fraction of 1% Bi atoms, substituting As atoms in the layer, predominantly enhances the spin-orbit coupling strength in its valence band. The presence of bismuth results in a small decrease in the ferromagnetic Curie temperature and a distinct increase in the coercive fields. On the other hand, the Bi incorporation into the layer strongly enhances the magnitude of negative magnetoresistance without affecting the hole concentration in the layer. The negative magnetoresistance is interpreted in terms of the suppression of weak localization in a magnetic field. Application of the weak-localization theory for two-dimensional ferromagnets by Dugaev et al. to the experimental magnetoresistance results indicates that the decrease in spin-orbit scattering length accounts for the enhanced magnetoresistance in (Ga,Mn)(Bi,As).

Theoretical Study of Dynamical and Electronic Properties of Noncentrosymmetric Superconductor NbReSi

The noncentrosymmetric NbReSi superconductor with Tc≃6.5 K is characterized by the relatively large upper critical magnetic field. Its multigap features were observed experimentally. Recent studies suggested the realization of P6¯2m or Ima2 symmetry. We discuss the dynamical properties of both symmetries (e.g., phonon spectra). In this paper, using the ab initio techniques, we clarify this ambiguity, and conclude that the Ima2 symmetry is unstable, and P6¯2m should be realized. The P6¯2m symmetry is also stable in the presence of external hydrostatic pressure. We show that NbReSi with the P6¯2m symmetry should host phonon surface states for (100) and (110) surfaces. Additionally, we discuss the main electronic properties of the system with the stable symmetry.

Fine Structure and the Huge Zero-Field Splitting in Ni2+ Complexes

We perform a thorough study of the ground state magnetic properties of nickel-based 3d8 complexes. This includes an in-depth analysis of the contribution of the crystal field, spin exchange and spin-orbit interactions to the ground state magnetic properties. Of particular interest to the current investigation are the presence and occurrence of non-trivial zero-field splitting. The study focuses on the cases of Ni2+ ideal octahedral, trigonal bipyramidal, square planar and tetrahedral geometries. We provide results for the complete energy spectrum, the fine structure related to the ground state and the second set of excited states, low-field magnetic susceptibility and magnetization. In addition, we examine the zero-field fine structure in square pyramidal, trigonal pyramidal and trigonal planar complexes. The obtained results unequivocally show that a moderate or highly coordinated 3d8 complex can neither exhibit spin-orbit-driven large and giant magnetic anisotropy nor a huge zero-field splitting. Moreover, in the trigonal bipyramidal coordination, a fine structure associated to the ground state cannot result from the spin-orbit coupling alone.

Effective potentials in a rotating spin-orbit-coupled spin-1 spinor condensate

We theoretically study the stationary-state vortex lattice configurations of rotating spin-orbit (SO)- and coherently-coupled spin-1 Bose-Einstein condensates (BECs) trapped in quasi-two-dimensional harmonic potentials. The combined effects of rotation, SO and coherent couplings are analyzed systematically from the single-particle perspective. Through the single-particle Hamiltonian, which is exactly solvable for one-dimensional coupling, we illustrate that a boson in these rotating SO- and coherently-coupled condensates are subjected to effective toroidal, symmetric double-well, or asymmetric double-well potentials under specific coupling and rotation strengths. In the presence of mean-field interactions, using the coupled Gross-Pitaevskii formalism at moderate to high rotation frequencies, the analytically obtained effective potential minima and the numerically obtained coarse-grained density maxima position are in excellent agreement. On rapid rotation, we further find that the spin-expectation per particle of an antiferromagnetic spin-1 BEC approaches unity indicating a similarity in the response with ferromagnetic SO-coupled condensates.

Spin-orbit coupling, orbitally entangled antiferromagnetic order, and collective spin-orbital excitations inSr2VO4

With electron fillingn = 1 in theSr2VO4compound, the octahedrally coordinatedt2gorbitals are strongly active when the tetragonal distortion induced crystal field is tuned by external agent such as pressure. Considering the full range of crystal field induced tetragonal splitting in a realistic three-orbital model, collective spin-orbital excitations are investigated using the generalized self-consistent plus fluctuation approach. At ambient pressure, an entangled orbital + antiferromagnetic order is found to be stabilized beyond a critical value (∼30 meV) of spin-orbit coupling which is in the realistic range for 3d ions. The behavior of the calculated energy scales of collective excitations with crystal field is consistent with that of the transition temperatures with pressure as obtained from susceptibility and resistivity anomalies in high-pressure studies.

A spin modulating device, tuned by the Fermi energy, in honeycomb-like substrates periodically stubbed with transition-metal-dichalkogenides

We investigate spin transport through graphene-like substrates stubbed vertically with transition-metal-dichalcogenides (TMDs). A tight-binding model is used based on a graphene-like Hamiltonian that includes different types of spin-orbit coupling (SOC) terms permitted by theC_3vsymmetry group in TMDs/graphene-like heterostructures. The results show a spin modulation obtained by tuning the strength and sign of the Fermi energyE_Fand not by varying the SOC strength as is mainly the case of Datta and Das. The spin conductance is directly controlled by the value ofE_F. In addition, a perfect electron-spin modulation is obtained when a vertical strain is introduced. In this case, the spin conductance exhibits a strong energy dependence. The results may open the route to a combination of graphene-like substrates with TMD stubs and the development of spin-transistor devices controlled by the Fermi energy rather than the SOC strength.

Totally Spin-Polarized Currents in an Interferometer with Spin-Orbit Coupling and the Absence of Magnetic Field Effects

The paper studies the electronic current in a one-dimensional lead under the effect of spin-orbit coupling and its injection into a metallic conductor through two contacts, forming a closed loop. When an external potential is applied, the time reversal symmetry is broken and the wave vector k of the circulating electrons that contribute to the current is spin-dependent. As the wave function phase depends upon the vector k, the closed path in the circuit produces spin-dependent current interference. This creates a physical scenario in which a spin-polarized current emerges, even in the absence of external magnetic fields or magnetic materials. It is possible to find points in the system's parameter space and, depending upon its geometry, the value of the Fermi energy and the spin-orbit intensities, for which the electronic states participating in the current have only one spin, creating a high and totally spin-polarized conductance. For a potential of a few tens of meV, it is possible to obtain a spin-polarized current of the order of μA. The properties of the obtained electronic current qualify the proposed device as a potentially important tool for spintronics applications.

Successive phase transitions of the spin-orbit-coupled metal Cd2Re2O7probed by high-resolution synchrotron x-ray diffraction

The 5dpyrochlore oxide superconductor Cd₂Re₂O₇(CRO) has attracted significant interest as a spin-orbit-coupled metal (SOCM) that spontaneously undergoes a phase transition to an odd-parity multipole phase by breaking the spatial inversion symmetry due to the Fermi liquid instability caused by strong spin-orbit coupling. Despite the significance of structural information during the transition, previous experimental results regarding lattice deformation have been elusive. We have conducted ultra-high resolution synchrotron radiation x-ray diffraction experiments on a high-quality CRO single crystal. The temperature-dependent splitting of the 0 0 16 and 0 0 14 reflections, which are allowed and forbidden, respectively, in the high-temperature cubic phase I (space groupFd-3m), has been clearly observed and reveals the following significant facts: inversion symmetry breaking and tetragonal distortion occur simultaneously atT_s1= 201.5(1) K; the previously believed first-order transition between phase II (I-4m2) and phase III (I4₁22) atT_s2∼120 K consists of two close second-order transitions atT_s2= 115.4(1) K andT_s3∼ 100 K; there is a new orthorhombic phase XI (F222) in between. The order parameters (OPs) of these continuous transitions are uniquely represented by a two-dimensional irreducible representationE_uof theO_hpoint group, and the OPs of phase XI are a linear combination of those of phases II and III. Each phase is believed to correspond to a distinct odd-parity multipole order, and the complex successive transitions observed may be the result of an electronic phase transition that resolves the Fermi liquid instability in the SOCM.

Tunable phase transitions in half-Heusler TbPtBi compound

We report various phase transitions in half-Heusler TbPtBi compound using density functional theory. Specifically, the inclusion of spin-orbit coupling (SOC) leads to the band inversion resulting in the transition from the metallic to the topological semimetallic phase. However, in the presence of SOC, there is a phase transition from the topological semimetal to the trivial semimetal when the material is subjected to compressive strain-7%. Subsequently, under the further increase of compressive strain(⩾​​-7%), we find an opening of a direct band gap at the point, driving the system from the trivial semimetallic to a semiconducting state with changes in the sequence of the bands. In the absence of SOC, only the transition from the metallic to the semiconducting phase is noticed. Under tensile strain, the TbPtBi compound maintains its phase as in the unstrained condition but with an increase in the hole pocket at the Fermi level, both in the absence and presence of SOC. These tunable phase transitions (especially as a fraction of strain) make this compound very promising for application in various quantum devices, such as highly sensitive strain gauges.

A Theoretical Investigation into the Homo- and Hetero-leptic Cu(I) Phosphorescent Complexes Bearing 2,9-dimethyl-1,10-phenanthroline and bis [2-(diphenylphosphino)phenyl]ether Ligand

Cu(I) complexes have received widespread attention as a promising alternative to traditional noble-metal complexes. Herein, we systematically study the properties of Cu(I) complexes from homo- to hetero-ligands, and found the following: (1) hetero-ligands are beneficial to regulate phosphorescent efficiency; (2) when the hetero-ligands in a tetracoordinated Cu(I) complex are 1:1, the ligands coordinate along the d_x2-y2 direction of Cu(I) ion, which can observably suppress structural deformation; (3) unlike the P^P ligand, the N^N ligand can enhance the participation of Cu(I) during the transition process; (4) the addition of an appropriate amount of P^P ligand can effectively raise the energy level of HOMO (highest occupied molecular orbital), enhance the proportion of LLCT (ligand-ligand charge transfer), and thereby increase the available singlet emission transition moments which can be borrowed, thus promoting the radiative decay process. As a result, this work provides a detailed understanding of the effects of different ligands in Cu(I) complexes, and provides a valuable reference and theoretical basis for regulating and designing the phosphorescent properties of Cu(I) complexes in the future.

Cr2XTe4(X = Si, Ge) monolayers: a new type of two-dimensional high-TCIsing ferromagnetic semiconductors with a large magnetic anisotropy

Two-dimensional (2D) ferromagnetic semiconductor (FMS) provides the ideal platform for the development of quantum information technology in nanoscale devices. However, most of them suffer from low Curie temperature and small magnetic anisotropic energy (MAE), severely limiting their practical application. In this work, by using first-principles calculations, we predicted two stable 2D materials, namely, Cr₂SiTe₄and Cr₂GeTe₄monolayers. Interestingly, both of them are intrinsic direct band gap FMSs (∼1 eV) with a large magnetization (8µ_Bf.u.^-1) and sizable MAE (∼500μ_eV Cr^-1). Monte Carlo simulations based on Heisenberg model suggest markedly high Curie temperatures of these monolayers (∼200 K). Besides, their high mechanical, dynamical, and thermal stabilities are further verified by elastic constants, phonon dispersion calculations, andab initiomolecular dynamics simulations. The outstanding attributes render Cr₂XTe₄(X = Si, Ge) monolayers broadening the candidates of 2D FMS for a wide range of applications.

Magnetoelectric effects in Josephson junctions

The review is devoted to the fundamental aspects and characteristic features of the magnetoelectric effects, reported in the literature on Josephson junctions (JJs). The main focus of the review is on the manifestations of the direct and inverse magnetoelectric effects in various types of Josephson systems. They provide a coupling of the magnetization in superconductor/ferromagnet/superconductor JJs to the Josephson current. The direct magnetoelectric effect is a driving force of spin torques acting on the ferromagnet inside the JJ. Therefore it is of key importance for the electrical control of the magnetization. The inverse magnetoelectric effect accounts for the back action of the magnetization dynamics on the Josephson subsystem, in particular, making the JJ to be in the resistive state in the presence of the magnetization dynamics of any origin. The perspectives of the coupling of the magnetization in JJs with ferromagnetic interlayers to the Josephson current via the magnetoelectric effects are discussed.

Effects of a single impurity in a Luttinger liquid with spin-orbit coupling

In quasi-1D conducting nanowires spin-orbit coupling destructs spin-charge separation, intrinsic to Tomonaga-Luttinger liquid (TLL). We study renormalization of a single scattering impurity in a such liquid. Performing bosonization of low-energy excitations and exploiting perturbative renormalization analysis we extend the phase portrait inK_σ-K_ρspace, obtained previously for TLL with decoupled spin-charge channels.

Experimental and computational approaches to study the high temperature thermoelectric properties of novel topological semimetal CoSi

Here, we study the thermoelectric properties of topological semimetal CoSi in the temperature range 300-800 K by using combined experimental and density functional theory (DFT) based methods. CoSi is synthesized using arc melting technique and the Rietveld refinement gives the lattice parameters ofa=b=c= 4.445 Å. The measured values of Seebeck coefficient (S) shows the non-monotonic behaviour in the studied temperature range with the value of ∼-81μV K^-1at room temperature. The |S| first increases till 560 K (∼-93μV K^-1) and then decreases up to 800 K (∼-84μV K^-1) indicating the dominating n-type behaviour in the full temperature range. The electrical conductivity,σ(thermal conductivity,κ) shows the monotonic decreasing (increasing) behaviour with the values of∼5.2×105(12.1 W m^-1 K^-1) and∼3.6×105(14.2 W m^-1 K^-1) Ω^-1 m^-1at 300 K and 800 K, respectively. Theκexhibits the temperature dependency as,κ∝T^0.16. The DFT based Boltzmann transport theory is used to understand these behaviour. The multi-band electron and hole pockets appear to be mainly responsible for deciding the temperature dependent transport behaviour. Specifically, the decrease in the |S| above 560 K and change in the slope ofσaround 450 K are due to the contribution of thermally generated charge carriers from the hole pockets. The temperature dependent relaxation time (τ) is computed by comparing the experimentalσwith calculatedσ/τand it shows temperature dependency of 1/T^0.35. Further this value ofτis used to calculate the temperature dependent electronic part of thermal conductivity (κ_e) and it gives a fairly good match with the experiment. Present study suggests that electronic band-structure obtained from DFT provides a reasonably good estimate of the transport coefficients of CoSi in the high temperature region of 300-800 K.

Topological Hall effect in SrRuO3thin films and heterostructures

Transition metal oxides hold a wide spectrum of fascinating properties endowed by the strong electron correlations. In 4dand 5doxides, exotic phases can be realized with the involvement of strong spin-orbit coupling (SOC), such as unconventional magnetism and topological superconductivity. Recently, topological Hall effects (THEs) and magnetic skyrmions have been uncovered in SrRuO₃thin films and heterostructures, where the presence of SOC and inversion symmetry breaking at the interface are believed to play a key role. Realization of magnetic skyrmions in oxides not only offers a platform to study topological physics with correlated electrons, but also opens up new possibilities for magnetic oxides using in the low-power spintronic devices. In this review, we discuss recent observations of THE and skyrmions in the SRO film interfaced with various materials, with a focus on the electric tuning of THE. We conclude with a discussion on the directions of future research in this field.

SOiCI and iCISO: combining iterative configuration interaction with spin-orbit coupling in two ways

The near-exact iCIPT2 approach for strongly correlated systems of electrons, which stems from the combination of iterative configuration interaction (iCI, an exact solver of full CI) with configuration selection for static correlation and second-order perturbation theory (PT2) for dynamic correlation, is extended to the relativistic domain. In the spirit of spin separation, relativistic effects are treated in two steps: scalar relativity is treated by the infinite-order, spin-free part of the exact two-component (X2C) relativistic Hamiltonian, whereas spin-orbit coupling (SOC) is treated by the first-order, Douglas-Kroll-Hess-like SOC operator derived from the same X2C Hamiltonian. Two possible combinations of iCIPT2 with SOC are considered, i.e., SOiCI and iCISO. The former treats SOC and electron correlation on an equal footing, whereas the latter treats SOC in the spirit of state interaction, by constructing and diagonalizing an effective spin-orbit Hamiltonian matrix in a small number of correlated scalar states. Both double group and time reversal symmetries are incorporated to simplify the computation. Pilot applications reveal that SOiCI is very accurate for the spin-orbit splitting (SOS) of heavy atoms, whereas the computationally very cheap iCISO can safely be applied to the SOS of light atoms and even of systems containing heavy atoms when SOC is largely quenched by ligand fields.

Spin-dependent band-gap formation for theL-gap surface state on the(22×3)reconstructed Au(111) surface

We elucidate how the free-electron-like energy dispersion of theL-gap surface state on a Au(111)-(1 × 1) surface is modified by the experimentally observed uniaxial reconstruction of the topmost atomic layer. For this purpose, we perform a first-principles embedded Green's function calculation for the(22×3)reconstructed semi-infinite Au(111) surface. The obtained band structure unfolded into the surface Brillouin zone of the (1 × 1) surface can be understood in terms of two spin-split parabolic bands centered at theΓ¯point, their umklapp-induced replicas centered at reciprocal lattice vectors of the superlattice (SL) with much weaker intensities, and mini band gaps at the crossing of two of them. More importantly, it is revealed that the band-gap size depends not only on the amplitude of the SL potential but also on mutual spin orientations of two crossing bands. Furthermore, we demonstrate that the band-gap size and the charge density distribution of the surface states are closely correlated with spatial profile of the SL potential.

Spin accumulation and spin Hall effect in a two-layer system with a thin ferromagnetic layer

Spin accumulation and spin current are phenomena that enhance the functionality of the devices operating with charge and spin. We calculated them for the system consisting of a ferroelectric barrier and a thin ferromagnetic layer when the current flows parallel to the interface. We assume Dresselhaus and Rashba spin-orbit coupling linear in electron wave number. We demonstrate that spin accumulation and spin current can be manipulated by changing the direction of the magnetization of the FM layer with respect to the crystallographic axes of the ferroelectric barrier.

Recent developments on the magnetic and electrical transport properties of FeRh- and Rh-based heterostructures

It is fascinating how the binary alloy FeRh has been the subject of a vast number of studies almost solely for a single-phase transition. This is, however, reasonable, considering how various degrees of freedom are intertwined around this phase transition. Furthermore, the tunability of this phase transition-the large response to tuning parameters, such as electric field and strain-endows FeRh huge potential in applications. Compared to the bulk counterpart, FeRh in the thin-film form is superior in many aspects: materials in thin-film form are often more technologically relevant in the first place; in addition, the substrates add extra dimensions to the tunability, especially when the substrate itself is multiferroic. Here we review recent developments on the magnetic and transport properties of heterostructures based on FeRh and its end-member Rh, with the latter providing a new route to exploiting spin-orbit interactions in functional spintronic heterostructures other than the more often employed 5dmetals. The methods utilized in the investigation of the physical properties in these systems, and the design principles employed in the engineering thereof may conceivably be extended to similar phase transitions to other magnetic materials.

Seebeck effect in nanomagnets

We present a theory of the Seebeck effect in nanomagnets with dimensions smaller than the spin diffusion length, showing that the spin accumulation generated by a temperature gradient strongly affects the thermopower. We also identify a correction arising from the transverse temperature gradient induced by the anomalous Ettingshausen effect and an induced spin-heat accumulation gradient. The relevance of these effects for nanoscale magnets is illustrated byab initiocalculations on dilute magnetic alloys.

Searching strong 'spin'-orbit coupled one-dimensional hole gas in strong magnetic fields

We show that a strong 'spin'-orbit coupled one-dimensional hole gas is achievable via applying a strong magnetic field to the original two-fold degenerate (spin degeneracy) hole gas confined in a cylindrical Ge nanowire. Both strong longitudinal and strong transverse magnetic fields are feasible to achieve this goal. Based on quasi-degenerate perturbation calculations, we show the induced low-energy subband dispersion of the hole gas can be written asE=ℏ2kz2/(2mh*)+ασzkz+gh*μBBσx/2, a form exactly the same as that of the electron gas in the conduction band. Here the Pauli matricesσ^z,xrepresent a pseudo spin (or 'spin'), because the real spin degree of freedom has been split off from the subband dispersions by the strong magnetic field. Also, for a moderate nanowire radiusR= 10 nm, the induced effective hole massmh*(0.065∼0.08me)and the 'spin'-orbit couplingα(0.35 ∼ 0.8 eV Å) have a small magnetic field dependence in the studied magnetic field interval 1 <B< 15 T, while the effectiveg-factorgh*of the hole 'spin' only has a small magnetic field dependence in the large field region.

External tuning of topological phase transitions induced by interaction driven mass renormalization

Electron-electron interactions can be useful for realizing new nontrivial topological phases of matter. Here, we show by means of a tight-binding model and mean field theory how electron-electron interactions can lead to a topological phase transition. By externally adding or removing electrons from the system a band inversion between two bands with different parity is induced. This leads to a topological nontrivial phase if spin-orbit coupling is present. Besides the toy-model illustrating this mechanism, we also propose SmB₆as a possible playground for experimentally realizing a topological phase transition by external tuning.

Theory of unidirectional magnetoresistance and nonlinear Hall effect

We study the unidirectional magnetoresistance (UMR) and the nonlinear Hall effect (NLHE) in the ferromagnetic Rashba model. For this purpose we derive expressions to describe the response of the electric current quadratic in the applied electric field. We compare two different formalisms, namely the standard Keldysh nonequilibrium formalism and the Moyal-Keldysh formalism, to derive the nonlinear conductivities of UMR and NLHE. We find that both formalisms lead to identical numerical results when applied to the ferromagnetic Rashba model. The UMR and the NLHE nonlinear conductivities tend to be comparable in magnitude according to our calculations. Additionally, their dependencies on the Rashba parameter and on the quasiparticle broadening are similar. The nonlinear zero-frequency response considered here is several orders of magnitude higher than the one at optical frequencies that describes the photocurrent generation in the ferromagnetic Rashba model. Additionally, we compare our Keldysh nonequilibrium expression in the independent-particle approximation to literature expressions of the UMR that have been obtained within the constant relaxation time approximation of the Boltzmann formalism. We find that both formalisms converge to the same analytical formula in the limit of infinite relaxation time. However, remarkably, we find that the Boltzmann result does not correspond to the intraband term of the Keldysh expression. Instead, the Boltzmann result corresponds to the sum of the intraband term and an interband term that can be brought into the form of an effective intraband term due to thef-sum rule.

Coexisting unconventional Rashba- and Zeeman-type spin splitting in Pb-adsorbed monolayer WSe2

Based on first-principles calculations, the unconventional Rashba- and Zeeman-type spin splitting can simultaneously coexist in the Pb-adsorbed monolayer WSe₂system. The first two adsorption configurationst₁andt₂show remarkable features under the spin-orbit coupling, in which two split energy branches show same spin states at the left or right side of Γ, and the spin polarization is reversed for both Rashba band branches. For the second adsorption configuration, an energy gap was observed near the unconventional spin polarization caused by the repelled Rashba bands for avoid crossing, and this gap can produce non-dissipative spin current by applying the voltage. The results fort₂configuration with spin reversal show that the repel band gap and Rashba parameter can be effectively regulated within the biaxial strain range of -8% to 6%. By changing the adsorption distancedbetween Pb and the neighboring Se atom layer, the reduceddcaused the transfer from Rashba-type to Zeeman-type spin splitting. This predicted adsorption system would be promising for spintronic applications.

Tunable Planar Hall Effect in (Ga,Mn)(Bi,As) Epitaxial Layers

We have thoroughly investigated the planar Hall effect (PHE) in the epitaxial layers of the quaternary compound (Ga,Mn)(Bi,As). The addition of a small amount of heavy Bi atoms to the prototype dilute ferromagnetic semiconductor (Ga,Mn)As enhances significantly the spin-orbit coupling strength in its valence band, which essentially modifies certain magnetoelectric properties of the material. Our investigations demonstrate that an addition of just 1% Bi atomic fraction, substituting As atoms in the (Ga,Mn)As crystal lattice, causes an increase in the PHE magnitude by a factor of 2.5. Moreover, Bi incorporation into the layers strongly enhances their coercive fields and uniaxial magneto-crystalline anisotropy between the in-plane ⟨110⟩ crystallographic directions in the layers grown under a compressive misfit strain. The displayed two-state behaviour of the PHE resistivity at zero magnetic field, which may be tuned by the control of applied field orientation, could be useful for application in spintronic devices, such as nonvolatile memory elements.

Spatial order in a two-dimensional spin-orbit-coupled spin-1/2 condensate: superlattice, multi-ring and stripe formation

We demonstrate the formation of stable spatially-ordered states in auniformand alsotrappedquasi-two-dimensional (quasi-2D) Rashba or Dresselhaus spin-orbit (SO) coupled pseudo spin-1/2 Bose-Einstein condensate using the mean-field Gross-Pitaevskii equation. For weak SO coupling, one can have a circularly-symmetric (0, +1)- or (0, -1)-type multi-ring state with intrinsic vorticity, for Rashba or Dresselhaus SO coupling, respectively, where the numbers in the parentheses denote the net angular momentum projection in the two components, in addition to a circularly-asymmetric degenerate state with zero net angular momentum projection. For intermediate SO couplings, in addition to the above two types, one can also have states with stripe pattern in component densities with no periodic modulation in total density. The stripe state continues to exist for large SO coupling. In addition, a new spatially-periodic state appears in the uniform system: asuperlatticestate, possessing some properties of asupersolid, with a square-lattice pattern in component densities and also in total density. In a trapped system the superlattice state is slightly different with multi-ring pattern in component density and a square-lattice pattern in total density. For an equal mixture of Rashba and Dresselhaus SO couplings, in both uniform and trapped systems, only stripe states are found for all strengths of SO couplings. In a uniform system all these states are quasi-2D solitonic states.

Astatine Facing Janus: Halogen Bonding vs. Charge-Shift Bonding

The nature of halogen-bond interactions was scrutinized from the perspective of astatine, potentially the strongest halogen-bond donor atom. In addition to its remarkable electronic properties (e.g., its higher aromaticity compared to benzene), C₆At₆ can be involved as a halogen-bond donor and acceptor. Two-component relativistic calculations and quantum chemical topology analyses were performed on C₆At₆ and its complexes as well as on their iodinated analogues for comparative purposes. The relativistic spin–orbit interaction was used as a tool to disclose the bonding patterns and the mechanisms that contribute to halogen-bond interactions. Despite the stronger polarizability of astatine, halogen bonds formed by C₆At₆ can be comparable or weaker than those of C₆I₆. This unexpected finding comes from the charge-shift bonding character of the C–At bonds. Because charge-shift bonding is connected to the Pauli repulsion between the bonding σ electrons and the σ lone-pair of astatine, it weakens the astatine electrophilicity at its σ-hole (reducing the charge transfer contribution to halogen bonding). These two antinomic characters, charge-shift bonding and halogen bonding, can result in weaker At-mediated interactions than their iodinated counterparts.

First-principle investigation of all types of topological nodal lines in a realistic P63/mmc type titanium selenide

Topological nodal line (TNL) materials with one-dimensional band-crossing points (BCPs) exhibit interesting electronic characteristics and have special applications in electronic devices. Normally, based on the slopes of the crossing bands, the BCPs can be divided into two types, i.e., type I and type II nodal points. Based on the combination of the different types of nodal points, the nodal lines (NLs) can be divided into three categories: (i) type I NL, type II NL, and hybrid NL, these being formed by type I nodal points, type II nodal points, and type I and II nodal points, respectively. Compared with the large number of predicted type I NL materials, there are less type II and hybrid NL materials. In this study, it is predicted that P6₃/mmc type TiSe metal is a topological material which exhibits all types of NL states. Furthermore, the dynamic stability as well as the effect of spin-orbit coupling on the topological signatures are examined. Also, the nontrivial surface states are shown to provide evidence for the occurrence of the NL states. This novel material can be seen as a good platform to use for further investigations on the three types of NLs and diverse fermions.

Supersolid-like states in a two-dimensional trapped spin-orbit-coupled spin-1 condensate

We study supersolid-like states in a quasi-two-dimensional trapped Rashba and Dresselhaus spin-orbit (SO) coupled spin-1 condensate. For small strengths of SO couplingγ(γ⪅ 0.75), in the ferromagnetic phase, circularly-symmetric (0, ±1, ±2)- and (∓1, 0, ±1)-type states are formed where the numbers in the parentheses denote the angular momentum of the vortex at the center of the components and where the upper (lower) sign correspond to Rashba (Dresselhaus) coupling; in the antiferromagnetic phase, only (∓1, 0, ±1)-type states are formed. For large strengths of SO coupling, supersolid-like superlattice and superstripe states are formed in the ferromagnetic phase. In the antiferromagnetic phase, for large strengths of SO coupling, supersolid-like superstripe and multi-ring states are formed. For an equal mixture of Rashba and Dresselhaus SO couplings, only a superstripe state is found. All these states are found to be dynamically stable and hence accessible in an experiment and will enhance the fundamental understanding of crystallization onto radially periodic states in solids.