Spin-orbit Density Research Articles

Revisiting the global minimum of Au10 clusters.

This study employs high-level quantum chemical calculations to determine the global minimum structure of Au10 clusters definitively. Contrary to previous reports, coupled-cluster singles and doubles with perturbative triples [CCSD(T)] calculations with sizable quadruple-ζ basis sets incorporating the spin-orbit (SO) effect reveal that the planar 10.b structure is the true global minimum for Au10, not the three-dimensional 10.a structure. Two-component spin-orbit density functional theory calculations demonstrate that the SO effect is minimal for most Au10 isomers, except for the 10.f structure. A straightforward diagnostic tool is proposed for identifying Au cluster structures with strong spin-orbit coupling based on 6p orbital occupation. The calculated IR spectra based on Boltzmann averaging the six Au10 isomers show good agreement with recent experimental spectra although minor discrepancies are noted potentially due to interactions with Kr. The results suggest that the transition point to non-planar global minimum structures for Au clusters lies beyond Au10 but is nearby.

Hund’s three rules in actinide-containing superatoms with spin-orbit coupling calculations

The intriguing and challenge issue in magnetic superatoms is searching for the suitable candidates to validate the Hund’s rules. Here, early actinide elements (An: Ac, Th, Pa, U, Np, Pu, Am) whose 5f electrons may crossover the localization and delocalization characteristics have been chosen to alloy with Al atoms in designing magnetic An@Al12 superatoms. By doing the global minimum structure search and the spin-orbital coupling density functional theory calculations, we provide an original idea to give theoretical argument that Hund’s three rules are still applicable in superatoms, which can be related to the fillings of highly localized An-5f orbitals into large exchange-splitting 2 F superatom orbitals. Specifically, selective 5f sub-orbitals of several An dopants can exhibit a dual nature in superatomic bonding, i.e. partial 5f electrons of Pa, U and Pu are reactive whereas all 5f electrons of Np and Am are highly localized. The molecular orbital analyses, combined with the qualitative interpretation of the phenomenological superatom sub-shell model, address the intricate interplays between the structure symmetry, electronic structure, spin and orbital magnetic moments. These findings have important implications for understanding the bonding and magnetic behaviors of An-containing superatoms and pave the way for designing novel magnetic superatoms.

Current density functional framework for spin-orbit coupling: Extension to periodic systems.

Spin-orbit coupling induces a current density in the ground state, which consequently requires a generalization for meta-generalized gradient approximations. That is, the exchange-correlation energy has to be constructed as an explicit functional of the current density, and a generalized kinetic energy density has to be formed to satisfy theoretical constraints. Herein, we generalize our previously presented formalism of spin-orbit current density functional theory [Holzer et al., J. Chem. Phys. 157, 204102 (2022)] to non-magnetic and magnetic periodic systems of arbitrary dimension. In addition to the ground-state exchange-correlation potential, analytical derivatives such as geometry gradients and stress tensors are implemented. The importance of the current density is assessed for band gaps, lattice constants, magnetic transitions, and Rashba splittings. In the latter, the impact of the current density may be larger than the deviation between different density functional approximations.

Ab initio-based determination of lanthanoid-radical exchange as visualised by inelastic neutron scattering.

Magnetic exchange coupling can modulate the slow magnetic relaxation in single-molecule magnets. Despite this, elucidation of exchange coupling remains a significant challenge for the lanthanoid(iii) ions, both experimentally and computationally. In this work, the crystal field splitting and 4f-π exchange coupling in the erbium-semiquinonate complex [ErTp2dbsq] (Er-dbsq; Tp- = hydro-tris(1-pyrazolyl)borate, dbsqH2 = 3,5-di-tert-butyl-1,2-semiquinone) have been determined by inelastic neutron scattering (INS), magnetometry, and CASSCF-SO ab initio calculations. A related complex with a diamagnetic ligand, [ErTp2trop] (Er-trop; tropH = tropolone), has been used as a model for the crystal field splitting in the absence of coupling. Magnetic and INS data indicate antiferromagnetic exchange for Er-dbsq with a coupling constant of Jex = -0.23 meV (-1.8 cm-1) (-2Jex formalism) and good agreement is found between theory and experiment, with the low energy magnetic and spectroscopic properties well modelled. Most notable is the ability of the ab initio modelling to reproduce the signature of interference between localised 4f states and delocalised π-radical states that is evident in the Q-dependence of the exchange excitation. This work highlights the power of combining INS with EPR and magnetometry for determination of ground state properties, as well as the enhanced capability of CASSCF-SO ab initio calculations and purposely developed ab initio-based theoretical models. We deliver an unprecedentedly detailed representation of the entangled character of 4f-π exchange states, which is obtained via an accurate image of the spin-orbital transition density between the 4f-π exchange coupled wavefunctions.

Ligand-Induced Symmetry Breaking as the Origin of Multiexponential Photoluminescence Decay in CdSe Quantum Dots.

The bright photoluminescence (PL) of colloidal CdSe quantum dots (QDs) makes them interesting for optical applications. For most of them, well-defined PL properties, dominated by a single excitonic state, are required. However, in many PL experiments with QD ensembles, multiexponential decay was observed. On the basis of spin-orbit density functional theory and screened configuration interaction calculations, we show that highly symmetric and defect-free CdSe QDs with diameters of 1.7 and 2.0 nm possess a multiexponential low-temperature PL at the single-dot level. This is a consequence of ligand-induced symmetry breaking with a subsequent rearrangement of the lowest eight excitonic states in two sets of four singly degenerate excitonic states. For each set, the lowest state is dark and the other three are bright. We find that the splitting between the sets can be modified by the coverage and choice of the ligand, which facilitates the engineering of the PL properties of CdSe QDs.

How Does Bending the Uranyl Unit Influence Its Spectroscopy and Luminescence?

Bent uranyl complexes can be formed with chloride ligands and 1,10-phenanthroline (phen) ligands bound to the equatorial and axial planes of the uranyl(VI) moiety, as revealed by the crystal structures, IR and Raman spectroscopy, and quantum-chemical calculations. With the goal of probing the influence of chloride and phenanthroline coordination enforcing the bending on the absorption and emission spectra of this complex, spin-orbit time-dependent density functional theory calculations for the bare uranyl complexes as well as for the free UO2Cl2 subunit and the UO2Cl2(phen)2 complex were performed. The emission spectra have been fully simulated by ab initio methods and compared to experimental photoluminescence spectra, recorded for the first time for UO2Cl2(phen)2. Notably, the bending of uranyl in UO2Cl2 and UO2Cl2(phen)2 triggers excitations of the uranyl bending mode, yielding a denser luminescence spectrum.

Periodic trends in trivalent actinide halides, phosphates, and arsenates.

Due to the limited abundance of the actinide elements, computational methods, for now, remain an exclusive avenue to investigate the periodic trends across the actinide series. As every actinide element can exhibit a +3-oxidation state, we have explored model systems of gas-phase actinide trihalides, phosphates, and arsenates across the series to capture the periodic trends. By doing so, we were able to capture the periodic trends down the halogen series as well, and for the first time we are reporting a study on actinide astatides. Using scalar and spin-orbit relativistic Density Functional Theory (DFT) calculations, we have explored the variations in bond lengths, bond angles, and the charges on actinides (An). Despite the use of different sets of ligands, the trends remain similar. The properties of trivalent Pa, U, Np, and Pu are nearly identical; similar ionic radii could be the reason. The actinide elements show a tendency to exhibit a pre-Pu and a post-Cm behaviour, with Am acting as a switch. This could be due to the change in the behaviour from d-f-type to f-filling/d-type at around Pu-Cm in the actinides as already proposed in the previous literature. Bond lengths in the AnX3 increase down the halide series, and the atomic charges decrease on the actinide elements.

Relativistic Spin-Orbit Electronegativity and the Chemical Bond Between a Heavy Atom and a Light Atom.

Relativistic effects are known to alter the chemical bonds and spectroscopic properties of heavy-element compounds. In this work, we introduce the concept of spin-orbit (SO) electronegativity of a heavy atom, as reflected by an SO-induced change in the interatomic distance between the heavy atom (HA) and a neighboring light atom (LA). We provide a transparent interpretation of these SO effects by using the concept of spin-orbit electron deformation density (SO-EDD). Spin-orbit coupling at the HA induces rearrangement of the electron density for the scalar-relativistically optimized geometry that, in turn, exerts a new force on the LA. The resulting expansion or contraction of the HA-LA bond depends on the nature and electron configuration of the HA. In addition, we quantify the change in atomic electronegativity induced by SO coupling for a series of hydrides, thereby complementing the SO-EDD picture. The trends in the SO-induced electronegativity and the HA-LA bond length across the periodic table of elements are demonstrated and interpreted, and also linked, intuitively, with the SO-induced NMR shielding at the LA.

Nuclear charge densities in spherical and deformed nuclei: Toward precise calculations of charge radii

Background: Precise measurements of atomic transitions affected by electron-nucleus hyperfine interactions offer sensitivity to explore basic properties of the atomic nucleus and study fundamental symmetries, including the search for new physics beyond the standard model of particle physics. In particular, such measurements, augmented by atomic and nuclear calculations, will enable extraction of the higher-order radial moments of the charge-density distribution in spherical and deformed nuclei. The new data impose higher precision requirements on a theoretical description.Purpose: The nuclear charge density is composed of the proton point distribution folded with the nucleonic charge distributions. The latter induce subtle relativistic corrections due to the coupling of nucleon magnetic moments with the nuclear spin-orbit density. Additional corrections come from the effect of center-of-mass projection. We assess the precision of nuclear charge density calculations by studying the behavior of relativistic and center-of-mass motion corrections to the second and fourth charge radial moments. Special attention has been paid to the magnetic spin-orbit density associated with the local variations of the spin-orbit current.Methods: The calculations for semimagic and open-shell nuclei are performed in the framework of self-consistent mean-field theory using quantified energy density functionals and density-dependent pairing forces. We used the general expression for the spin-orbit form factor that is valid for spherical and deformed nuclei.Results: We studied the impact of various correction terms on the charge radii, fourth radial moments, diffraction radii, and surface thickness of spherical and deformed nuclei. The spin-orbit corrections to charge radial moments and surface thickness show strong shell fluctuations which can make an appreciable effect when aiming at high-precision predictions of isotopic shifts. The inclusion of relativistic and center-of-mass corrections impacts the quality of energy density functionals optimized to charge radii data.Conclusions: To establish reliable constraints on the existence of new forces from isotope shift measurements, precise calculations of nuclear charge densities of deformed nuclei are needed. The proper inclusion of the spin-orbit charge density and other correction terms is essential when aiming at extraction of subtle effects which become particularly visible in isotopic trends. It is also important when developing high-quality nuclear energy density functionals optimized using heterogeneous datasets involving absolute charge radii, differential charge radii, and charge form factor properties deduced from electron-scattering data.

4f -block elemental-atom-embedded gh−C3N4 monolayers: Large magnetic moment, high-temperature ferromagnetism, and huge magnetic anisotropy energy

Two-dimensional (2D) magnetic materials with strong magnetism, long magnetic relaxation times, high temperature ferromagnetism, and large magnetic anisotropy energy (MAE) are promising for applications of nanoscale spintronic devices. Using the spin-orbital coupling density functional theory (DFT) calculations, we investigate the structural stability, electronic structures, and magnetic properties of graphenelike carbon-nitride ($\mathrm{gh}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$) sheets with the adsorption of whole series of $4f$-block elemental (lanthanide; Ln) atoms. Our results demonstrate that Ln atoms can be stably embedded into/above the center of the cavity of $\mathrm{gh}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$ monolayer and significantly affect the electronic and magnetic properties. Upon single Ln atom adsorption, all Ln@$\mathrm{gh}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$ systems show metal character, large spin and orbital magnetic moments, and most of them favor the long-range ferromagnetic ordering. Interestingly, Pr, Nd, Ho adsorbates would possess total magnetic moments of 1.09, 1.47, 6.13 ${\ensuremath{\mu}}_{B}$, high Curie temperatures (${T}_{c}$) of 593, 699, 700 K, large MAEs of 4.89, $\ensuremath{-}17.88$, 21.31 meV/Ln atom, respectively. Based on the electronic structure analyses, we propose that the $4f$ electron hopping between the occupied and unoccupied states around the Fermi level contributes to the large MAE, significant intratomic Ln-$sd$ orbital hybridization with N-$2p$ orbital hybridization gives rise to structural stability, and the coexistence of superexchange and RKKY interactions determines the long-term ferromagnetic coupling between Ln atoms. The present study demonstrates that Ln@$\mathrm{gh}\text{\ensuremath{-}}{\mathrm{C}}_{3}{\mathrm{N}}_{4}$ sheets have significant promise for applications in spintronics such as high density memory devices or for magnetic random access memory.

Origin of the evolution of spin-orbit and pseudospin-orbit splittings in neutron drops

We found the same staggering pattern in the evolution of the spin-orbit (SO) and pseudospin-orbit (PSO) splittings in neutron drops obtained by the energy-density functional (EDF) theory using SAMi-T and SLy5 parametrizations, with and without the tensor terms respectively. This staggering evolution is similar to the results recently obtained by the relativistic Brueckner-Hartree-Fock (RBHF) theory with the Bonn A potential, which was claimed to be a tensor effect. In this work, we present an intuitive but essential explanation on the origin of this staggering evolution in the Skyrme EDF theory. We found that, for both the SO and PSO partners, the energy splittings mainly originate from the SO potential. The staggering evolution of both the SO and PSO splittings is mainly due to the drastic change of density while neutrons fill different single-particle orbits. The key to determine the specific staggering pattern is the strength of the neutron density gradient term and the SO density term in the SO potential, not necessarily the tensor force.

Electronic correlation effects in Pb quantum wires on Si(557)

The low-temperature spin-orbit density phase of Pb quantum wires on Si(557) was investigated by scanning tunneling microscopy. High resolution images of this phase allow the proposal of an atomistic model for the densely-packed Pb coverages on the (223) facets. A metal-insulator phase transition was confirmed by local spectroscopy and the BCS-like gap feature of $2\mathrm{\ensuremath{\Delta}}\ensuremath{\approx}30\phantom{\rule{0.28em}{0ex}}\text{meV}$ at 10 K correlates with the transition temperature of ${T}_{c}=78\phantom{\rule{0.28em}{0ex}}\text{K}$ seen in surface transport experiments. Moreover, Pb excess coverage adds an extra row of Pb atoms on the mini-(111) terraces. These atoms induce a metallic state located at the step edge at low temperature and quench the charge and spin correlations of the spin-orbit density phase.

Antiunitary symmetry protected higher-order topological phases

Higher-order topological (HOT) phases feature boundary (such as corner and hinge) modes of codimension $d_c>1$. We here identify an \emph{antiunitary} operator that ensures the spectral symmetry of a two-dimensional HOT insulator and the existence of cornered localized states ($d_c=2$) at precise zero energy. Such an antiunitary symmetry allows us to construct a generalized HOT insulator that continues to host corner modes even in the presence of a \emph{weak} anomalous Hall insulator and a spin-orbital density wave orderings, and is characterized by a quantized quadrupolar moment $Q_{xy}=0.5$. Similar conclusions can be drawn for the time-reversal symmetry breaking HOT $p+id$ superconductor and the corner localized Majorana zero modes survive even in the presence of weak Zeeman coupling and $s$-wave pairing. Such HOT insulators also serve as the building blocks of three-dimensional second-order Weyl semimetals, supporting one-dimensional hinge modes.

Spin-Orbital Density Wave and a Mott Insulator in a Two-Orbital Hubbard Model on a Honeycomb Lattice

Inspired by the recent discovery of correlated insulating states in twisted bilayer graphene, we study a two-orbital Hubbard model on the honeycomb lattice with two electrons per unit cell. Based on the real-space density matrix renormalization group simulation, we identify a metal-insulator transition around U_{c}/t=2.5-3. In the vicinity of U_{c}, we find strong spin-orbital density wave fluctuations at commensurate wave vectors, accompanied by weaker incommensurate charge density wave fluctuations. The spin-orbital density wave fluctuations are enhanced with increasing system sizes, suggesting the possible emergence of long-range order in the two-dimensional limit. At larger U, our calculations indicate a possible nonmagnetic Mott insulator phase without spin or orbital polarization. Our findings offer new insight into correlated electron phenomena in twisted bilayer graphene and other multiorbital honeycomb materials.

Ab initio calculations for the H-decorated neutral and charged oxygen vacancy in erbium oxide

Point defects created by neutron/electron irradiation often determine the performance and the chemical stability and activity of oxide materials. Oxygen vacancies constitute a common point defect in metal oxides, such as erbium oxide (Er2O3), one of the candidate materials for tritium permeation barriers in fusion blanket systems. However, a systematic study of the oxygen vacancy properties in Er2O3 has been lacking. Here, the properties of isolated neutral and charged oxygen vacancies in Er2O3 are investigated by means of supercell total-energy calculations using a first-principles method based on density–functional theory. Vacancy formation energies, hydrogen–vacancy interactions, and geometry rearrangements around these point defects are investigated in detail. The characterization of the electronic structure of these point defects is established by the analysis of the spin-orbital density of states. It is found that the energetic and electronic properties of the oxygen vacancies depend on the chemical potentials of the O and Er atoms in Er2O3. The defect formation energy decreases when one hydrogen atom is trapped into the vacancy in both charged oxidizing and reduced environments. Also, the interstitial hydrogen 1s orbital forms multicenter bonds with the 5d orbital of the neighboring Er atoms, and the net charge transfer from the defected Er2O3 is 1.04e, 1.02e, and 0.80e for the defect charge states of 1−, 0 and 1+, respectively. We suggest that the H-decorated vacancy defect may be responsible for the increase of the self-healing properties of Er2O3.

Nitrogen Reduction by Multimetallic trans-Uranium Actinide Complexes: A Theoretical Comparison of Np and Pu to U.

There is recent interest in organometallic complexes of the trans-uranium elements. However, preparation and characterization of such complexes are hampered by radioactivity and chemotoxicity issues as well as the air-sensitive and poorly understood behavior of existing compounds. As such, there are no examples of small-molecule activation via redox reactivity of organometallic trans-uranium complexes. This contrasts with the situation for uranium. Indeed, a multimetallic uranium(III) nitride complex was recently synthesized, characterized, and shown to be able to capture and functionalize molecular nitrogen (N2) through a four-electron reduction process, N2 → N24-. The bis-uranium nitride, U-N-U core of this complex is held in a potassium siloxide framework. Importantly, the N24- product could be further functionalized to yield ammonia (NH3) and other desirable species. Using the U-N-U potassium siloxide complex, K3U-N-U, and its cesium analogue, Cs3U-N-U, as starting points, we use scalar-relativistic and spin-orbit coupled density functional theory calculations to shed light on the energetics and mechanism for N2 capture and functionalization. The N2 → N24- reactivity depends on the redox potentials of the U(III) centers and crucially on the stability of the starting complex with respect to decomposition into the mixed oxidation U(IV)/U(III) K2U-N-U or Cs2U-N-U species. For the trans-uranium, Np and Pu analogues of K3U-N-U, the N2 → N24- process is endoergic and would not occur. Interestingly, modification of the Np-O and Pu-O bonds between the actinide cores and the coordinated siloxide framework to Np-NH, Pu-NH, Np-CH2, and Pu-CH2 bonds drastically improves the reaction free energies. The Np-NH species are stable and can reductively capture and reduce N2 to N24-. This is supported by analysis of the spin densities, molecular structure, long-range dispersion effects, as well as spin-orbit coupling effects. These findings chart a path for achieving small-molecule activation with organometallic neptunium analogues of existing uranium complexes.

Excited-State Reactivity of [Mn(im)(CO)3 (phen)]+ : A Structural Exploration.

The electronic excited state reactivity of [Mn(im)(CO)3 (phen)]+ (phen = 1,10-phenanthroline; im = imidazole) ranging between 420 and 330 nm have been analyzed by means of relativistic spin-orbit time-dependent density functional theory and wavefunction approaches (state-average-complete-active-space self-consistent-field/multistate CAS second-order perturbation theory). Minimum energy conical intersection (MECI) structures and connecting pathways were explored using the artificial force induced reaction (AFIR) method. MECIs between the first and second singlet excited states (S1 /S2 -MECIs) were searched by the single-component AFIR (SC-AFIR) algorithm combined with the gradient projection type optimizer. The structural, electronic, and excited states properties of [Mn(im)(CO)3 (phen)]+ are compared to those of the Re(I) analogue [Re(im)(CO)3 (phen)]+ . The high density of excited states and the presence of low-lying metal-centered states that characterize the Mn complex add complexity to the photophysics and open various dissociative channels for both the CO and imidazole ligands. © 2018 Wiley Periodicals, Inc.

Photophysical Heavy-Atom Effect in Iodinated Metallocorroles: Spin-Orbit Coupling and Density of States.

Excited-state dynamics and electronic structures of Al and Ga corrole complexes were studied as a function of the number of β-pyrrole iodine substituents. Using spectrally broad-band femtosecond-resolved fluorescence upconversion, we determined the kinetics of the Soret fluorescence decay, the concomitant rise and subsequent decay of the Q-band fluorescence, as well as of the accompanying vibrational relaxation. Iodination was found to accelerate all involved processes. The time constant of the internal conversion from the Soret to the Q states decreases from 320-540 to 70-185 fs upon iodination. Vibrational relaxation then occurs with about 15 and 0.36-1.4 ps lifetime for iodine-free and iodinated complexes, respectively. Intersystem crossing to the lowest triplet is accelerated up to 200 times from nanoseconds to 15-24 ps; its rate correlates with the iodine p(π) participation in the corrole π-system and the spin-orbit coupling (SOC) strength. TDDFT calculations with explicit SOC show that iodination introduces a manifold of low-lying singlet and triplet iodine → corrole charge-transfer (CT) states. These states affect the photophysics by (i) providing a relaxation cascade for the Soret → Q internal conversion and cooling and (ii) opening new SOC pathways whereby CT triplet character is admixed into both Q singlet excited states. In addition, SOC between the higher Q singlet and the Soret triplet is enhanced as the iodine participation in frontier corrole π-orbitals increases. Our observations that iodination of the chromophore periphery affects the whole photocycle by changing the electronic structure, spin-orbit coupling, and the density of states rationalize the "heavy-atom effect" and have implications for controlling excited-state dynamics in a range of triplet photosensitizers.

Magnetic ground state of the multiferroic hexagonal LuFeO3

The structural, electric, and magnetic properties of bulk hexagonal $\mathrm{LuFe}{\mathrm{O}}_{3}$ are investigated. Single phase hexagonal $\mathrm{LuFe}{\mathrm{O}}_{3}$ has been successfully stabilized in the bulk form without any doping by sol-gel method. The hexagonal crystal structure with $P{6}_{3}cm$ space group has been confirmed by x-ray-diffraction, neutron-diffraction, and Raman spectroscopy study at room temperature. Neutron diffraction confirms the hexagonal phase of $\mathrm{LuFe}{\mathrm{O}}_{3}$ persists down to 6 K. Further, the x-ray photoelectron spectroscopy established the 3+ oxidation state of Fe ions. The temperature-dependent magnetic dc susceptibility, specific heat, and neutron-diffraction studies confirm an antiferromagnetic ordering below the N\'eel temperature $({T}_{\mathrm{N}})\ensuremath{\sim}130\phantom{\rule{0.16em}{0ex}}\mathrm{K}$. Analysis of magnetic neutron-diffraction patterns reveals an in-plane ($ab$-plane) ${120}^{\ensuremath{\circ}}$ antiferromagnetic structure, characterized by a propagation vector $k=(0\phantom{\rule{0.16em}{0ex}}0\phantom{\rule{0.16em}{0ex}}0)$ with an ordered moment of $2.84\phantom{\rule{0.16em}{0ex}}{\ensuremath{\mu}}_{\mathrm{B}}/\mathrm{F}{\mathrm{e}}^{3+}$ at 6 K. The ${120}^{\ensuremath{\circ}}$ antifferomagnetic ordering is further confirmed by spin-orbit coupling density functional theory calculations. The on-site coulomb interaction ($U$) and Hund's parameter $({J}_{H})$ on Fe atoms reproduced the neutron-diffraction ${\mathrm{\ensuremath{\Gamma}}}_{1}$ spin pattern among the Fe atoms. $P\text{\ensuremath{-}}E$ loop measurements at room temperature confirm an intrinsic ferroelectricity of the sample with remnant polarization ${P}_{\mathrm{r}}\ensuremath{\sim}0.18\phantom{\rule{0.16em}{0ex}}\ensuremath{\mu}\mathrm{C}/\mathrm{c}{\mathrm{m}}^{2}$. A clear anomaly in the dielectric data is observed at $\ensuremath{\sim}{T}_{\mathrm{N}}$ revealing the presence of magnetoelectric coupling. A change in the lattice constants at ${T}_{\mathrm{N}}$ has also been found, indicating the presence of a strong magnetoelastic coupling. Thus a coupling between lattice, electric, and magnetic degrees of freedom is established in bulk hexagonal $\mathrm{LuFe}{\mathrm{O}}_{3}$.

Correlation Effects and Hidden Spin-Orbit Entangled Electronic Order in Parent and Electron-Doped Iridates Sr2IrO4

Analogs of the high-T$_c$ cuprates have been long sought after in transition metal oxides. Due to the strong spin-orbit coupling (SOC), the $5d$ perovskite iridates Sr$_2$IrO$_4$ exhibit a low-energy electronic structure remarkably similar to the cuprates. Whether a superconducting state exists as in the cuprates requires understanding the correlated spin-orbit entangled electronic states. Recent experiments discovered hidden order in the parent and electron doped iridates, some with striking analogies to the cuprates, including Fermi surface pockets, Fermi arcs, and pseudogap. Here, we study the correlation and disorder effects in a five-orbital model derived from the band theory. We find that the experimental observations are consistent with a $d$-wave spin-orbit density wave order that breaks the symmetry of a joint two-fold spin-orbital rotation followed by a lattice translation. There is a Berry phase and a plaquette spin flux due to spin procession as electrons hop between Ir atoms, akin to the intersite SOC in quantum spin Hall insulators. The associated staggered circulating $J_\text{eff}=1/2$ spin current can be probed by advanced techniques of spin-current detection in spintronics. This electronic order can emerge spontaneously from the intersite Coulomb interactions between the spatially extended iridium $5d$ orbitals, turning the metallic state into an electron doped quasi-2D Dirac semimetal with important implications on the possible superconducting state suggested by recent experiments.