An anomalous optical second-harmonic generation (SHG) signal was previously reported in Sr$_2$IrO$_4$ and attributed to a hidden odd-parity bulk magnetic state. Here we investigate the origin of this SHG signal using a combination of bulk magnetic susceptibility, magnetic-field-dependent SHG rotational anisotropy, and overlapping wide-field SHG imaging and atomic force microscopy measurements. We find that the anomalous SHG signal exhibits a two-fold rotational symmetry as a function of in-plane magnetic field orientation that is associated with a crystallographic distortion. We also show a change in SHG signal across step edges that tracks the bulk antiferromagnetic stacking pattern. While we do not rule out the existence of hidden order in Sr$_2$IrO$_4$, our results altogether show that the anomalous SHG signal in parent Sr$_2$IrO$_4$ originates instead from a surface-magnetization-induced electric-dipole process that is enhanced by strong spin-orbit coupling.
We report a global structural distortion in Sr$_2$IrO$_4$ using spatially resolved optical second and third harmonic generation rotational anisotropy measurements. A symmetry lowering from an $I4_{1}/acd$ to $I4_{1}/a$ space group is observed both above and below the Néel temperature that arises from a staggered tetragonal distortion of the oxygen octahedra. By studying an effective super-exchange Hamiltonian that accounts for this lowered symmetry, we find that perfect locking between the octahedral rotation and magnetic moment canting angles can persist even in the presence of large non-cubi…
The fractionalization of elementary excitations in quantum spin systems is a central theme in current condensed matter physics. The Kitaev honeycomb spin model provides a prominent example of exotic fractionalized quasiparticles, composed of itinerant Majorana fermions and gapped gauge fluxes. However, identification of the Majorana fermions in a three-dimensional honeycomb lattice remains elusive. Here we report spectroscopic signatures of fractional excitations in the harmonic-honeycomb iridates $β$- and $γ$-Li$_2$IrO$_3$. Using polarization resolved Raman spectroscopy, we find that the dyna…
By combining 5d transition-metal oxides showing pronounced spin-orbit interactions and oxide-based heterostructures, we propose rutile-based IrO2/TiO2 superlattices as promising candidates for unconventional electronic properties. By means of density-functional-theory simulations complemented with Hubbard-like corrections, we focus on the evolution of the electronic structure as a function of the IrO2 layer thickness and predict the heterostructures to exhibit a thickness-controlled metal-to-insulator transition, crucially related to the connectivity among IrO6 octahedra. The subtle interplay …
A three-orbital itinerant-electron model involving d_{xz}, d_{yz} and d_{xy} Fe 3d orbitals is proposed for iron pnictides towards understanding the (π,0) ordered magnetism and magnetic excitations in these materials. It is shown that this model at half filling yields a gapped (π,0) magnetic state with high degree of robustness and stability, and simultaneously reproduces several experimentally observed features such as the electronic structure, spin excitations, as well as the ferro orbital order between the d_{xz} and d_{yz} orbitals.
We report equilibrium and nonequilibrium optical measurements on the recently synthesized "harmonic" honeycomb iridate γ-Li$_2$IrO$_3$ (LIO), as well as the layered honeycomb iridate Na$_2$IrO$_3$ (NIO). Using Fourier transform infrared microscopy we performed reflectance measurements on LIO, from which we obtained the optical conductivity below 2 eV. In addition we measured the photoinduced changed in reflectance, ΔR, as a function of time, t, temperature, T, and probe field polarization in both LIO and NIO. In LIO, ΔR(t,T) is anisotropic and comprised of three T dependent components. Two of …
We present a microscopic model for the anisotropic exchange interactions in Sr$_{2}$IrO$_{4}$. A direct construction of Wannier functions from first-principles calculations proves the $j_{\mathrm{eff}}$=1/2 character of the spin-orbit integrated states at the Fermi level. An effective $j_{\mathrm{eff}}$-spin Hamiltonian explains the observed weak ferromagnetism and anisotropy of antiferromagnetically ordered magnetic state, which arise naturally from the $j_{\mathrm{eff}}$=1/2 state with a rotation of IrO$_{6}$ octahedra. It is suggested that Sr$_{2}$IrO$_{4}$ is a unique class of materials wi…
Resonant x-ray diffraction experiments were performed for the metallic iridium oxide IrO$_{2}$. We observed anisotropic tensor of susceptibility (ATS) scattering, the spectrum of which shows a sharp contrast between the $L_{3}$ and $L_{2}$ edges. At the $L_{3}$ edge, resonance excitations were clearly observed from the core 2$p$ orbitals to both the 5$d$ $t_{2g}$ and $e_{g}$ orbitals. In contrast, the resonance mode associated with 5$d$ $t_{2g}$ orbitals was indiscernible at the $L_{2}$ edge. This contrasting behavior indicates that Ir 5$d$ $t_{2g}$ orbitals are fairly close to the $J_{\rm eff…
To examine the insulating mechanism of 5$d$ transition metal oxide Sr$_2$IrO$_4$, we study the ground state properties of a three-orbital Hubbard model with a large relativistic spin-orbit coupling on a square lattice. Using a variational Monte Carlo method, we find that the insulating state appearing in the ground state phase diagram for one hole per site varies from a weakly correlated to a strongly correlated antiferromagnetic (AF) state with increasing Coulomb interactions. This crossover is characterized by the different energy gain mechanisms of the AF insulating state, i.e., from an int…
{\it Ab initio} analyses of A$_2$IrO$_4$ (A=Sr, Ba) are presented. Effective Hubbard-type models for Ir 5$d$ $t_{2g}$ manifolds downfolded from the global band structure are solved based on the dynamical mean-field theory. The results for A=Sr and Ba correctly reproduce paramagnetic metals undergoing continuous transitions to insulators below the Néel temperature $T_N$. These compounds are classified not into Mott insulators but into Slater insulators. However, the insulating gap opens by a synergy of the Neel order and significant band renormalization, which is also manifested by a 2D bad met…
The infrared transmission measurement was performed for a two-dimensional spin-orbit coupling induced Mott insulator Sr$_{2}$IrO$_{4}$ with a layered-perovskite structure. A phonon-assisted magnon mode was observed at around 1870 cm$^{-1}$ in the absorption spectrum below 180 K. The nearest-neighbor superexchange interaction $J$ was estimated to be 57 meV, which is about a half of the $J$ value of a copper oxide La$_{2}$CuO$_{4}$. Our results indicate the possibility that nontrivial properties such as superconductivity are realized at rather high temperature in iridium oxides.
We have studied in detail the electronic structure of IrO$_2$ including spin-orbit coupling (SOC) and electron-electron interaction, both within the GGA+U and GGA+DMFT approximations. Our calculations reveal that the Ir t$_{2g}$ states at the Fermi level largely retain the J$_{\rm eff}$ = $\frac{1}{2}$ character, suggesting that this complex spin-orbit entangled state may be robust even in metallic IrO$_2$. We have calculated the phase diagram for the ground state of IrO$_2$ as a function of $U$ and find a metal insulator transition that coincides with a magnetic phase change, where the effect…
A family of insulating iridates with chemical formula Li$_2$IrO$_3$ has recently been discovered, featuring three distinct crystal structures $α,β,γ$ (honeycomb, hyperhoneycomb, stripyhoneycomb). Measurements on the three-dimensional polytypes, $β$- and $γ$-Li$_2$IrO$_3$, found that they magnetically order into remarkably similar spiral phases, exhibiting a non-coplanar counter-rotating spiral magnetic order with equivalent q=0.57 wavevectors. We examine magnetic Hamiltonians for this family and show that the same triplet of nearest-neighbor Kitaev-Heisenberg-Ising (KJI) interactions reproduce…
Spin wave excitations and stability of the ($π,0$) ordered magnetic state are investigated in a minimal two-band itinerant-electron model for iron pnictides. Presence of hopping anisotropy generates a strong ferro-orbital order in the $d_{xz}$ and $d_{yz}$ Fe orbitals. The orbital order sign is as observed in experiments. The induced ferro-orbital order strongly enhances the spin wave energy scale and stabilizes the magnetic state by optimizing the strength of the emergent AF and F spin couplings through optimal band fillings in the two orbitals. The calculated spin-wave dispersion is in quant…
Hyperhoneycomb iridate $β$-Li$_2$IrO$_3$ is a three-dimensional analogue of two-dimensional honeycomb iridates, such as $α$-Li$_2$IrO$_3$, which recently appeared as another playground for the physics of Kitaev-type spin liquid. $β$-Li$_2$IrO$_3$ shows a non-collinear spiral ordering of spin-orbital-entangled $J_{\rm eff}$ = 1/2 moments at low temperature, which is known to be suppressed under a pressure of $\sim$2 GPa. With further increase of pressure, a structural transition is observed at $P_{\rm S}$ $\sim$ 4 GPa at room temperature. Using the neutron powder diffraction technique, the crys…
We report a high-field electron spin resonance study in the sub-THz frequency domain of a single crystal of Sr$_2$IrO$_4$ that has been recently proposed as a prototypical spin-orbital Mott insulator. In the antiferromagnetically (AFM) ordered state with noncollinear spin structure that occurs in this material at $T_{\rm N} \approx 240$ K we observe both the "low" frequency mode due to the precession of weak ferromagnetic moments arising from a spin canting, and the "high" frequency modes due to the precession of the AFM sublattices. Surprisingly, the energy gap for the AFM excitations appears…
We report the existence of a phase transition at high temperature in the 3D Kitaev candidate material, $β$-Li$_2$IrO$_3$. We show that the transition is bulk, intrinsic and orders a tiny magnetic moment with a spatially anisotropic saturation moment. We show that even though this transition is global, it does not freeze the local Ir moments, which order at much lower temperatures into an incommensurate state. Rather, the ordered moment has an orbital origin that is coupled to spin correlations, likely of a Kitaev origin. The separate ordering of spin-correlated orbital moments and of local Ir …
Aimed at identifying the role of microscopic Hamiltonian parameters on spin wave excitations, orbital order, and magnetic moments in the $(π,0)$ SDW state, two different three band models for iron pnictides are compared --- one at two-third filling ($n=4$) and another at half filling $(n=3)$ --- both of which yield qualitatively correct Fermi surface structure in the paramagnetic state. Spin wave analysis of the model at $n=4$ shows instability of the SDW state, which is attributed to weakly developed magnetic moments and weak magnetic couplings due to overfilling, as inferred from the observe…
We present an anti-ferromagnetically ordered ground state of Na$_{2}$IrO$_{3}$ based on density-functional-theory calculations including both spin-orbit coupling and on-site Coulomb interaction $U$. We show that the splitting of $e_{g}'$ doublet states by the strong spin-orbit coupling is mainly responsible for the intriguing nature of its insulating gap and magnetic ground state. Due to its proximity to the spin-orbit insulator phase, the magnetic ordering as obtained with finite $U$ is found to exhibit a strong in-plane anisotropy. The phase diagram of Na$_{2}$IrO$_{3}$ suggests a possible i…
In this paper, we study theoretically the behaviors of the magnetic excitations(MEs) in the normal and nematic phases of iron pnictides. The normal state MEs exhibit commensurability to diamond and to square-like structure transition with the increase of energy. This structure transition persists in the spin and orbital scenarios of nematic phases, although the MEs show anisotropic behaviors due to the C4 symmetry breaking induced by the nematic orders. The MEs exhibit distinct energy evolution behaviors between the spin and orbital scenarios of nematicity. For the spinnematic scenario, the an…
A Dirac fermion emerges as a result of interplay between symmetry and topology in condensed matter. Current research moves towards investigating the Dirac fermions in the presence of manybody effects in correlated system. Here, we demonstrate the emergence of correlation-induced symmetry-protected Dirac semimetal state in the lightly-doped spin-orbit-coupled Mott insulator Sr$_2$IrO$_4$. We find that the nonsymmorphic crystalline symmetry stabilizes a Dirac line-node semimetal and that the correlation-induced symmetry-breaking electronic order further leads to a phase transition from the Dirac…
CaBaFe4O7 is a mixed-valent transition metal oxide having both Fe2+ and Fe3+ ions in tetrahedral coordination. Here we characterize its magnetic properties by magnetization measurements and investigate its local electronic structure using soft x-ray absorption spectroscopy at the Fe L2,3 edges, in combination with multiplet cluster and spin-resolved band structure calculations. We found that the Fe2+ ion in the unusual tetrahedral coordination is Jahn-Teller active with the high-spin e^2 (up) t2^3 (up) e^1 (down) configuration having a x^2-y^2-like electron for the minority spin. We deduce tha…
A number of recent experiments exhibit electronic anisotropy in the iron pnictides, and there is a growing body of experimental evidence that its origin is related to orbital ordering in Fe d_{xz} and d_{yz} orbitals. We examine this problem in the parent compounds of the iron pnictides by a combination of ab initio band theory calculations, phenomenological Ginzburg-Landau theory of coupled orbital and magnetic order parameters, and a microscopic mean-field study of the Kugel-Khomskii model. We find that orbital ordering is sufficient to explain a number of key experimental observations, in p…
The system of an iron phthalocyanine molecule on the Au(111) surface, has been studied recently due to its peculiar properties. In particular, several surprising results of scanning tunneling spectroscopy changing the position of the molecule and applying magnetic field can be explained by the {\it non-Landau} Fermi liquid state of a 2-channel spin-1 Kondo model with anisotropy. The localized orbitals near the Fermi level are three, one of symmetry $z^2$ and two (nearly) degenerate $π$ orbitals of symmetry $xz$ and $yz$. Previous studies using the numerical renormalization group neglected one …
We report a resonant inelastic x-ray scattering (RIXS) investigation of ultra-thin epitaxial films of Ba$_2$IrO$_4$, and compare their low energy magnetic and spin-orbit excitations to those of their sister compound Sr$_2$IrO$_4$. Due to the 180$^\circ$ Ir-O-Ir bond, the bandwidth of the magnon and spin-orbiton is significantly larger in Ba$_2$IrO$_4$, making it difficult to describe these two types of excitations as separate well-defined quasiparticles. Both types of excitations are found to be quite sensitive to the effect of epitaxial strain. In addition, we find that the d-level inversion …
The underdoped phase diagram of the iron-based superconductors exemplifies the complexity common to many correlated materials. Indeed, multiple ordered states that break different symmetries but display comparable transition temperatures are present. Here, we argue that such a complexity can be understood within a simple unifying framework. This framework, built to respect the symmetries of the non-symmorphic space group of the FeAs/Se layer, consists of primary magnetically-ordered states and their vestigial phases that intertwine spin and orbital degrees of freedom. All vestigial phases have…
The so-called neutron spin resonance consists of a prominent enhancement of the magnetic response at a particular energy and momentum transfer upon entering the superconducting state of unconventional superconductors. In the case of iron-based superconductors, the neutron resonance has been extensively studied experimentally, and a peculiar spin-space anisotropy has been identified by polarized inelastic neutron scattering experiments. Here we perform a theoretical study of the energy- and spin-resolved magnetic susceptibility in the superconducting state with $ s_{+-} $-wave order parameter, …
We investigate the electronic structure and magnetic properties of a $J_\text{eff} = 1/2$ iridate Ca$_4$IrO$_6$ and the implications of doping electrons and holes using ab initio density functional theory. Our calculations considering spin-orbit interaction reveal that although the Mott-insulating parent compound transforms into a conductor upon doping, antiferromagnetism sustains in the doped system, albeit with a grossly noncollinear arrangement of the spins. We find a strong spin-orbit interaction and magneto-crystalline anisotropy, causing frustration in the system, possibly leading to the…
In most magnetically-ordered iron pnictides, the magnetic moments lie in the FeAs planes, parallel to the modulation direction of the spin stripes. However, recent experiments in hole-doped iron pnictides have observed a reorientation of the magnetic moments from in-plane to out-of-plane. Interestingly, this reorientation is accompanied by a change in the magnetic ground state from a stripe antiferromagnet to a tetragonal non-uniform magnetic configuration. Motivated by these recent observations, here we investigate the origin of the spin anisotropy in iron pnictides using an itinerant microsc…
We investigate the two-orbital Hubbard model, which reproduces the electron and hole Fermi surfaces in the iron pnictide superconductors, in the presence of the Jahn-Teller electron-phonon coupling by using the dynamical mean-field theory. When the intra- and inter-orbital Coulomb interactions, $U$ and $U'$, increase with $U=U'$, both the local spin and orbital susceptibilities, $χ_{s}$ and $χ_{o}$, increase with $χ_{s}=χ_{o}$ because of the spin-orbital symmetry. Due to the Hund's rule coupling $J$, $χ_{s}$ is enhanced and dominates over $χ_{o}$ resulting in the repulsive local pairing intera…
We show how an accurate first-principles treatment of the canted-antiferromagnetic ground state of Sr$_2$IrO$_4$, a prototypical $5d$ correlated spin-orbit coupled material, can be obtained without invoking any free parameters such as the Hubbard U or tuning the spin-orbit coupling strength. Our theoretically predicted iridium magnetic moment of 0.250 $μ_B$, canted by 12.6$^{\circ}$ off the a-axis, is in accord with experimental results. By resolving the magnetic moments into their spin and orbital components, we show that our theoretically obtained variation of the magnetic scattering amplitu…
Nematic order resulting from the partial melting of density-waves has been proposed as the mechanism to explain nematicity in iron-based superconductors. An outstanding question, however, is whether the microscopic electronic model for these systems -- the multi-orbital Hubbard model -- displays such an ordered state as its leading instability. In contrast to usual electronic instabilities, such as magnetic and charge order, this fluctuation-driven phenomenon cannot be captured by the standard RPA method. Here, by including fluctuations beyond RPA in the multi-orbital Hubbard model, we derive …
Recent experiments in iron pnictide superconductors reveal that, as the putative magnetic quantum critical point is approached, different types of magnetic order coexist over a narrow region of the phase diagram. Although these magnetic configurations share the same wave-vectors, they break distinct symmetries of the lattice. Importantly, the highest superconducting transition temperature takes place close to this proliferation of near-degenerate magnetic states. In this paper, we employ a renormalization group calculation to show that such a behavior naturally arises due to the effects of spi…
Spin-orbit coupling (SOC) is a fundamental interaction in solids which can induce a broad spectrum of unusual physical properties from topologically non-trivial insulating states to unconventional pairing in superconductors. In iron-based superconductors (IBS) its role has so far been considered insignificant with the models based on spin- or orbital fluctuations pairing being the most advanced in the field. Using angle-resolved photoemission spectroscopy we directly observe a sizeable spin-orbit splitting in all main families of IBS. We demonstrate that its impact on the low-energy electronic…
We show here that orbital degrees of freedom produce a distinct signature in the magnetic excitation spectrum of iron-based superconductors above the magnetic ordering temperature. Because $d_{xz}$ and $d_{yz}$ orbitals are strongly connected with the Fermi surface topology, the nature of magnetic excitations can be modified significantly due to the presence of either static or fluctuating orbital correlations. Within a five-orbital itinerant model, we show that static orbital order generally leads to an enhancement of commensurate magnetic excitations even when the original Fermi surface lack…
We study a two-orbital spin model to describe (pi,0) stripe antiferromagnetism in the iron pnictides. The "double-spin" model has an on-site Hunds's coupling and inter-site interactions extending to second neighbors (inter- and intra-orbital) on the square lattice. Using a variational method based on a cluster decomposition, we optimize wave functions with up to 8 cluster sites (up to 2^16 variational parameters). We focus on the anomalously small ordered moments in the stripe state of the pnictides. To account for it, and large variations among different compounds, we show that the second-nei…
We have investigated the electronic and optical properties of (Sr$_{1-x}$Ca$_{x}$)$_2$IrO$_4$ (x= 0 - 0.375) and (Sr$_{1-y}$Ba$_y$)$_2$IrO$_4$ (y= 0 - 0.375) epitaxial thin-films, in which the bandwidth is systematically tuned via chemical substitutions of Sr ions by Ca and Ba. Transport measurements indicate that the thin-film series exhibits insulating behavior, similar to the J$_{eff}$= 1/2 spin-orbit Mott insulator Sr$_2$IrO$_4$. As the average A-site ionic radius increases from (Sr$_{1-x}$Ca$_{x}$)$_2$IrO$_4$ to (Sr$_{1-y}$Ba$_y$)$_2$IrO$_4$, optical conductivity spectra in the near-infra…
We construct the symmetry adapted low energy effective Hamiltonian for the electronic states in the vicinity of the Fermi level in iron based superconductors. We use Luttinger's method of invariants, expanding about Gamma and M points in the Brillouin zone corresponding to two iron unit cell, and then matching the coefficients of the expansion to the 5- and 8-band models. We then use the method of invariants to study the effects of the spin-density wave order parameters on the electronic spectrum, with and without spin-orbit coupling included. Among the results of this analysis is the finding …
Electronic structure has been studied in lightly electron doped correlated spin-orbit insulator Sr$_2$IrO$_4$ by angle-resolved photoelectron spectroscopy. We have observed coexistence of the lower Hubbard band and the in-gap band, the momentum dependence of the latter traces that of the band calculations without on-site Coulomb repulsion. The in-gap state remained anisotropically gapped in all observed momentum area, forming a remnant Fermi surface state, evolving towards the Fermi energy by carrier doping. These experimental results show a striking similarity with those observed in deeply un…
The magnetic excitations of the iron pnictides are explained within a degenerate double-exchange model. The local-moment spins are coupled by superexchanges $J_1$ and $J_2$ between nearest and next-nearest neighbors, respectively, and interact with the itinerant electrons of the degenerate $d_{xz}$ and $d_{yz}$ orbitals via a ferromagnetic Hund exchange. The latter stabilizes $(π,0)$ stripe antiferromagnetism due to emergent ferro-orbital order and the resulting kinetic energy gain by hopping preferably along the ferromagnetic spin direction. Taking the quantum nature of the spins into account…
Here we show that the 2.80(8) μB/Fe block antiferromagnetic order of BaFe2Se3 transforms into stripe antiferromagnetic order in KFe2Se3 with a decrease in moment to 2.1(1) μB/Fe. This reduction is larger than expected from the change in electron count from Ba$^{2+}$ to K$^{+}$, and occurs with the loss of the displacements of Fe atoms from ideal positions in the ladders, as found by neutron pair distribution function analysis. Intermediate compositions remain insulating, and magnetic susceptibility measurements show a suppression of magnetic order and probable formation of a spin-glass. Togeth…
We examine the appearance of the experimentally-observed stripe spin-density-wave magnetic order in five different orbital models of the iron pnictide parent compounds. A restricted mean-field ansatz is used to determine the magnetic phase diagram of each model. Using the random phase approximation, we then check this phase diagram by evaluating the static spin susceptibility in the paramagnetic state close to the mean-field phase boundaries. The momenta for which the susceptibility is peaked indicate in an unbiased way the actual ordering vector of the nearby mean-field state. The dominant or…
Iridates are of considerable current interest because of the strong spin-orbit coupling that leads to a variety of new phenomena. Using density-functional studies, we predict the formation of a spin-orbital entangled two-dimensional electron gas (2DEG) in the $δ$-doped iridate La$_δ$Sr$_2$IrO$_4$, where a single SrO layer is replaced by a LaO layer. The extra La electron resides close to the $δ$-doped layer, partially occupying the $J_{\rm eff}= 1/2 $ upper Hubbard band and thereby making the interface metallic. The magnetic structure of the bulk is destroyed near the interface, with the Ir$_0…
The low-energy electronic states of the iron-based superconductors are strongly affected by both spin-orbit coupling and, when present, by the nematic order. These two effects have different physical origins, yet they can lead to similar gap features in the electronic spectrum. Here we show how to disentangle them experimentally in the iron superconductors with one Fe plane per unit cell. Although the splitting of the low energy doublet at the Brillouin zone center ($Γ$-point) can be due to either the spin-orbit coupling or the nematic order, or both, the degeneracy of each of the doublet stat…
