Continuum of magnetic excitations in the Kitaev honeycomb iridate D$_3$LiIr$_2$O$_6$
Thomas Halloran, Yishu Wang, K. W. Plumb, M. B. Stone, Barry Winn, M. K. Graves-Brook, J. A. Rodriguez-Rivera, Yiming Qui, Prashant Chauhan, Johannes Knolle, Ro… 展开作者
Thomas Halloran, Yishu Wang, K. W. Plumb, M. B. Stone, Barry Winn, M. K. Graves-Brook, J. A. Rodriguez-Rivera, Yiming Qui, Prashant Chauhan, Johannes Knolle, Roderich Moessner, N. P. Armitage, Tomohiro Takayama, Hidenori Takagi, Collin Broholm
Inelastic neutron scattering (INS) measurements of powder D$_3(^{7}$Li)($^{193}$Ir)$_2$O$_6$ reveal low energy magnetic excitations with a scattering cross section that is broad in $|Q|$ and consistent with a Kitaev spin-liquid (KSL) state. The magnetic nature of the excitation spectrum is demonstrated by longitudinally polarized neutron studies. The total magnetic moment of 1.7(2)$μ_B$/Ir inferred from the total magnetic scattering cross section is consistent with the effective moment inferred from magnetic susceptibility data and expectations for the $J_{\rm eff}=1/2$ single ion state. The rise in the dynamic correlation function ${\cal S}(Q,ω)$ for $\hbarω<5~$meV can be described by a nearest-neighbor Kitaev model with interaction strength $K\approx-13(5)$~meV. Exchange disorder associated with the mixed D-Li site could play an important role in stabilizing the low $T$ quantum fluctuating state.
We report low-temperature (down to $\sim$5 K) Raman signatures of the recently discovered intercalated honeycomb magnet Ag$_3$LiIr$_2$O$_6$, a putative Kitaev quantum spin liquid (QSL) candidate. The Kitaev QSL is predicted to host Majorana fermions as its emergent elementary excitations through a thermal fractionalization of entangled spins $S = 1/2$. We observe evidence of this fractionalization in the low-energy magnetic continuum whose temperature evolution harbours signatures of the predicted Fermi statistics obeyed by the itinerant Majorana quasiparticles. The magnetic Raman susceptibili…
Spin-orbit coupling (SOC) offers a large variety of novel and extraordinary magnetic and electronic properties in otherwise `ordinary pool' of heavy ion oxides. Here we present a detailed study on an apparently isolated hexagonal 2$H$ spin-chain $d^4$ iridate Sr$_3$LiIrO$_6$ (SLIO) with geometric frustration. Our structural studies clearly reveal perfect Li-Ir chemical order in this compound. Our combined experimental and {\it ab-initio} electronic structure investigations establish a magnetic ground state with finite Ir$^{5+}$ magnetic moments in this compound, contrary to the anticipated non…
The 5d-electron honeycomb compound H$_3$LiIr$_2$O$_3$ [K. Kitagawa $et$ $al.$, Nature 554, 341-345 (2018)] exhibits an apparent quantum spin liquid (QSL) state. In this intercalated spin-orbital compound, a remarkable pile up of low-energy states was experimentally observed in specific heat and nuclear magnetic (NMR) spin relaxation. We show that a bond disordered Kitaev model can naturally account for this phenomenon, suggesting that disorder plays an essential role in its theoretical description. In the exactly soluble Kitaev model, we obtain, via spin fractionalization, a random bipartite h…
H3LiIr2O6 is the first honeycomb-lattice system without any signs of long-range magnetic order down to the lowest temperatures, raising the hope for the realization of an ideal Kitaev quantum spin liquid. Its honeycomb layers are coupled by interlayer hydrogen bonds. Static or dynamic disorder of these hydrogen bonds was proposed to strongly affect the magnetic exchange and to make Kitaev-type interactions dominant. Using dielectric spectroscopy, here we provide experimental evidence for dipolar relaxations in H3LiIr2O6 and deuterated D3LiIr2O6, which mirror the dynamics of protons and deutero…
Recently H$_3$LiIr$_2$O$_6$ has been reported as a spin-orbital entangled quantum spin liquid (QSL) [K. Kitagawa et al., Nature {\bf 554}, 341 (2018)], albeit its connection to Kitaev QSL has not been yet identified. To unveil the related Kitaev physics, we perform the first Raman spectroscopy studies on single crystalline H$_3$LiIr$_2$O$_6$ samples. We implement a soft chemical replacement of Li$^+$ with H$^+$ from $α$-Li$_2$IrO$_3$ single crystals to synthesize the single crystal samples of the iridate second generation H$_3$LiIr$_2$O$_6$. The Raman spectroscopy can be used to diagnose the Q…
Very recently a quantum liquid was reported to form in H$_3$LiIr$_2$O$_6$, an iridate proposed to be a close realization of the Kitaev honeycomb model. To test this assertion we perform detailed quantum chemistry calculations to determine the magnetic interactions between Ir moments. We find that weakly bond dependent ferromagnetic Kitaev exchange dominates over other couplings, but still is substantially lower than in Na$_2$IrO$_3$. This reduction is caused by the peculiar position of the inter-layer species: removing hydrogen cations next to a Ir$_2$O$_2$ plaquette increases the Kitaev excha…
We use x-ray spectroscopy at Ir L$_3$/L$_2$ absorption edge to study powder samples of the intercalated honeycomb magnet Ag$_3$LiIr$_2$O$_6$. Based on x-ray absorption and resonant inelastic x-ray scattering measurements, and exact diagonalization calculations including next-neighbour Ir-Ir electron hoping integrals, we argue that the intercalation of Ag atoms results in a nearly itinerant electronic structure with enhanced Ir-O hybridization. As a result of the departure from the local relativistic $j_{\rm eff}\! = \!1/2$ state, we find that the relative orbital contribution to the magnetic m…
We investigate the structural and magnetic properties of a Kitaev spin liquid candidate material Ag$_3$LiIr$_2$O$_6$ based on $^7$Li nuclear magnetic resonance line shape, Knight shift and spin-lattice relaxation rate $1/T_1$. The first sample A shows signatures of magnetically ordered spins, and exhibits one sharp $^7$Li peak with FWHM increasing significantly below 14~K. $1/T_1^{stretch}$ of this sample displays a broad local maximum at 40~K, followed by a very sharp peak at $T_N = 9\pm1$~K due to critical slowing down of Ir spin fluctuations, a typical signature of magnetic long range order…
Searching for an ideal Kitaev spin liquid candidate with anyonic excitations and long-range entanglement has motivated the synthesis of a new family of intercalated Kitaev magnets such as H$_{3}$LiIr$_{2}$O$_{6}$, Cu$_{2}$IrO$_{3}$, and Ag$_{3}$LiIr$_{2}$O$_{6}$. The absence of a susceptibility peak and a two-step release of the magnetic entropy in these materials has been proposed as evidence of proximity to the Kitaev spin liquid. Here we present a comparative study of the magnetic susceptibility, heat capacity, and muon spin relaxation ($μ$SR) between two samples of Ag$_{3}$LiIr$_{2}$O$_{6}…
A new quantum spin liquid (QSL) candidate material H$_{3}$LiIr$_{2}$O$_{6}$ was synthesized recently and was found not to show any magnetic order or phase transition down to low temperatures. In this work, we study the quantum dynamics of the hydrogen ions, i.e., protons, in this material by combining first-principles calculations and theoretical analysis. We show that each proton and its adjacent oxygen ions form an electric dipole. The dipole interactions and the proton tunneling are captured by a transverse-field Ising model with a quantum disordered paraelectric ground state. The dipole ex…
Kitaev magnets are materials with bond-dependent Ising interactions between localized spins on a honeycomb lattice. Such interactions could lead to a quantum spin-liquid (QSL) ground state at zero temperature. Recent theoretical studies suggest two potential signatures of a QSL at finite temperatures, namely a scaling behavior of thermodynamic quantities in the presence of quenched disorder, and a two-step release of the magnetic entropy. Here, we present both signatures in Ag$_{3}$LiIr$_{2}$O$_{6}$ which is synthesized from $α$-Li$_{2}$IrO$_{3}$ by replacing the inter-layer Li atoms with Ag a…
We have investigated experimentally and theoretically the series (Na$_{1-x}$Li$_{x}$)$_{2}$IrO$_{3}$. Contrary to what has been believed so far, only for $x\leq0.25$ the system forms uniform solid solutions. For larger Li content, as evidenced by powder X-ray diffraction, scanning electron microscopy and density functional theory calculations, the system shows a miscibility gap and a phase separation into an ordered Na$_{3}$LiIr$_2$O$_{6}$ phase with alternating Na$_3$ and LiIr$_2$O$_6$ planes, and a Li-rich phase close to pure Li$_{2}$IrO$_{3}$. For $x\leq 0.25$ we observe (1) an increase of …
A Kitaev-Heisenberg-J2-J3 model is proposed to describe the Mott-insulating layered iridates A2IrO3 (A=Na,Li). The model is a combination of the Kitaev honeycomb model and the Heisenberg model with all three nearest neighbor couplings J1, J2 and J3. A rich phase diagram is obtained at the classical level, including the experimentally suggested zigzag ordered phase; as well as the stripy phase, which extends from the Kitaev-Heisenberg limit to the J1-J2-J3 one. Combining the experimentally observed spin order with the optimal fitting to the uniform magnetic susceptibility data gives an estimate…
We have performed inelastic neutron scattering experiments to investigate the magnetic excitations in the quasi-face-centered-cubic (fcc) iridate double perovskites La2ZnIrO6 and La2MgIrO6, which are characterized by A-type antiferromagnetic ground states. The neutron scattering data, which reveal gapped spin wave excitations with a very weak dispersion, are shown to be well-described by theoretical calculations on a model Hamiltonian with dominant Kitaev interactions. Our surprising finding shows that in contrast to honeycomb materials, where the Kitaev interaction manifests itself via comple…
The nature of magnetism in the intercalated honeycomb iridate Ag$_3$LiIr$_2$O$_6$ has been a subject of recent intensive debate, where the absence or presence of antiferromagnetic order has been reported to be related to possible structural disorder effects and, an enhanced Ir-O hybridization and itinerancy with respect to the parent Li$_2$IrO$_3$ has been suggested as the origin of distinct x-ray spectroscopy features. In the present work we investigate the microscopic nature of the electronic and magnetic properties of Ag$_3$LiIr$_2$O$_6$ via a combination of density functional theory combin…
Recently, realizations of Kitaev physics have been sought in the A2IrO3 family of honeycomb iridates, originating from oxygen-mediated exchange through edge-shared octahedra. However, for the j = 1/2 Mott insulator in these materials exchange from direct d-orbital overlap is relevant, and it was proposed that a Heisenberg term should be added to the Kitaev model. Here we provide the generic nearest-neighbour spin Hamiltonian when both oxygen-mediated and direct overlap are present, containing a bond dependent off-diagonal exchange in addition to Heisenberg and Kitaev terms. We analyze this com…
The Kitaev honeycomb model has attracted significant attention due to its exactly solvable spin-liquid ground state with fractionalized Majorana excitations and its possible materialization in magnetic Mott insulators with strong spin-orbit couplings. Recently, the 5d-electron compound H$_{3}$LiIr$_{2}$O$_{6}$ has shown to be a strong candidate for Kitaev physics considering the absence of any signs of a long-range ordered magnetic state. In this work, we demonstrate that a finite density of random vacancies in the Kitaev model gives rise to a striking pileup of low-energy Majorana eigenmodes …
The recently discovered three-dimensional hyperhoneycomb iridate, $β$-Li$_2$IrO$_3$, has raised hopes for the realization of dominant Kitaev interaction between spin-orbit entangled local moments due to its near-ideal lattice structure. If true, this material may lie close to the sought-after quantum spin liquid phase in three dimensions. Utilizing ab-initio electronic structure calculations, we first show that the spin-orbit entangled basis, $j_{\rm eff}$=1/2, correctly captures the low energy electronic structure. The effective spin model derived in the strong coupling limit supplemented by …
The unconventional magnetic orderings found in Kitaev spin liquid candidates suggest frustration of spin interactions, and raise a possibility of nearby spin liquid phases. In particular, counter-rotating spiral ordering in three-dimensional (3D) iridates is striking, and understanding the microscopic mechanism of such ordering may provide routes to 3D Kitaev spin liquids. We study a minimal 3D model including Kitaev $K$ and a symmetric off-diagonal bond-dependent $Γ$ interaction on the hyperhoneycomb lattice using exact diagonalization. We first show that a 12-site unitary transformation unve…
The magnetic field response of the Mott-insulating honeycomb iridate Na$_{2}$IrO$_{3}$ is investigated using torque magnetometry measurements in magnetic fields up to 60 tesla. A peak-dip structure is observed in the torque response at magnetic fields corresponding to an energy scale close to the zigzag ordering ($\approx 15~K$) temperature. Using exact diagonalization calculations, we show that such a distinctive signature in the torque response constrains the effective spin models for these classes of Kitaev materials to ones with dominant ferromagnetic Kitaev interactions, while alternative…
We present a thorough experimental study of the three-dimensional hyperhoneycomb Kitaev magnet $β$-Li$_2$IrO$_3$, using a combination of inelastic neutron scattering (INS), time-domain THz spectroscopy, and heat capacity measurements. The main results include a massive low-temperature reorganization of the INS spectral weight that evolves into a broad peak centered around 12 meV, and a distinctive peak in the THz data at 2.8(1) meV. A detailed comparison to powder-averaged spin-wave theory calculations reveals that the positions of these two features are controlled by the anisotropic $Γ$ coupl…
A decade ago, Alexei Kitaev proposed an exactly solvable $S$ = 1/2 model on a two-dimensional honeycomb lattice, where the spins fractionalize into Majorana fermions and form a topological quantum spin liquid (QSL) in the ground state. It was soon recognized that a family of complex iridium oxides, as well as ruthenium chloride, with honeycomb structure are magnetic insulators and accommodate essential ingredients of the Kitaev model, due to the interplay of electron correlation and spin-orbit coupling. This initiated a race to materialize the Kitaev QSL and to capture the signature of Majoran…
Combining Raman scattering measurements with mean field calculations of the Raman response we show that Kitaev-like magnetic exchange is dominant in the hyperkagome iridate Na$_4$Ir$_3$O$_8$. In the measurements we observe a broad Raman band at $\sim$~3500 cm$^{-1}$ with a band-width $\sim 1700$~cm$^{-1}$. Calculations of the Raman response of the Kitaev-Heisenberg model on the hyperkagome lattice shows that the experimental observations are consistent with calculated Raman response where Kitaev exchange interaction ($J_K$) is much larger than the Heisenberg term J$_1$ ($J_1/J_K \sim 0.1$). A …
We study theoretically the Raman scattering response $I(ω)$ in the gapless quantum spin liquid phase of the Kitaev-Heisenberg model. The dominant polarization-independent contribution $I_K (ω)$ reflects the density of states of the emergent Majorana fermions in the ground-state flux-sector. The integrability-breaking Heisenberg exchange generates a second contribution, whose dominant part $I_H (ω)$ has the form of a quantum quench corresponding to an abrupt insertion of four $Z_2$ gauge fluxes. This results in a weakly polarization dependent response with a sharp peak at the energy of the flux…
The realization of Kitaev spin liquid, where spins on a honeycomb lattice are coupled ferromagnetically by bond-dependent anisotropic interactions, has been a sought-after dream. 5d iridium oxides $α$-Li2IrO3 and $α$-Na2IrO3 with a honeycomb lattice of Jeff = 1/2 moments recently emerged as a possible materialization. Strong signature of Kitaev physics, however, was not captured. Here we report the discovery of a complex iridium oxide $β$-Li2IrO3 with Jeff = 1/2 moments on "hyper-honeycomb" lattice, a three-dimensional analogue of honeycomb lattice. A positive Curie-Weiss temperature $θ_{CW}$ …
Fifteen years since its inception, the Kitaev model still boasts only a narrow group of material realizations. We review the progress in studying and understanding one of them, lithium iridate Li$_2$IrO$_3$ available in three polymorphs that host strong Kitaev interactions on spin lattices of different dimensionality and topology. We also discuss feasibility, effectiveness, and repercussions of tuning strategies based on the application of external pressure and chemical substitutions.
The Kitaev-Heisenberg (KH) model has been proposed to capture magnetic interactions in iridate Mott insulators on the honeycomb lattice. We show that analogous interactions arise in many other geometries built from edge-sharing IrO_6 octahedra, including the pyrochlore and hyperkagome lattices relevant to Ir2O4 and Na4Ir3O8 respectively. The Kitaev spin liquid exact solution does not generalize to these lattices. However, a different exactly soluble point of the honeycomb lattice KH model, obtained by a four-sublattice transformation to a ferromagnet, generalizes to all these lattices. A Klein…
We study the nearest neighboring spin-1/2 Kitaev-Heisenberg-Gamma ($KJΓ$) model on the honeycomb lattice in the parameter region of ferromagnetic (FM) Kitaev and antiferromagnetic (AFM) Heisenberg couplings relevant for honeycomb iridates, using a coupled-chain analysis. Starting from the gapless Luttinger liquid phase of a decoupled $KJΓ$ chain, the inter-chain interactions in the two-dimensional model is treated within a self-consistent mean field approach based on the Luttinger liquid theory. In the FM Gamma region, our analysis recovers the reported 120$^\circ$ magnetic order, previously o…
The Kitaev honeycomb model has been suggested as a useful starting point to describe the honeycomb iridates. However, the zigzag ordering seen in Na2IrO3 and the magnetic transition in Li2IrO3 have raised questions to their connection to the Kitaev model and to the microscopic origin of these magnetic states, given their structural similarities. Here we study how the magnetic phases near the Kitaev spin liquid are affected by the inclusion of trigonal distortion of the oxygen octahedra within a purely nearest neighbour model. Using a combination classical analysis and exact diagonalization we …
In the quest for realizations of quantum spin liquids, the exploration of Kitaev materials - spin-orbit entangled Mott insulators with strong bond-directional exchanges - has taken center stage. However, in these materials the local spin-orbital j=1/2 moments typically show long-range magnetic order at low temperature, thus defying the formation of a spin-liquid ground state. Using resonant inelastic x-ray scattering (RIXS), we here report on a proximate spin liquid regime with clear fingerprints of Kitaev physics in the magnetic excitations of the honeycomb iridates alpha-Li2IrO3 and Na2IrO3.…
Quantum magnets with significant bond-directional Ising interactions, so-called Kitaev materials, have attracted tremendous attention recently in the search for exotic spin liquid states. Here we present a comprehensive set of measurements that enables us to investigate the crystal structures, Ir$^{4+}$ single ion properties, and magnetic ground states of the double perovskite iridates La$_2B$IrO$_6$ ($B$ $=$ Mg, Zn) and $A_2$CeIrO$_6$ ($A$ $=$ Ba, Sr) with a large nearest neighbor distance $>$ 5 Angstroms between Ir$^{4+}$ ions. Our neutron powder diffraction data on Ba$_2$CeIrO$_6$ can be re…
We have explored the hidden symmetries of a generic four-parameter nearest-neighbor spin model, allowed in honeycomb lattice compounds under trigonal compression. Our method utilizes a systematic algorithm to identify all dual transformations of the model that map the Hamiltonian on itself, changing the parameters and providing exact links between different points in its parameter space. We have found the complete set of points of hidden SU(2) symmetry at which seemingly highly anisotropic model can be mapped back on the Heisenberg model and inherits therefore its properties such as the presen…
The honeycomb iridates A2IrO3 (A=Na, Li) constitute promising candidate materials to realize the Heisenberg-Kitaev model (HKM) in nature, hosting unconventional magnetic as well as spin liquid phases. Recent experiments suggest, however, that Li2IrO3 exhibits a magnetically ordered state of incommensurate spiral type which has not been identified in the HKM. We show that these findings can be understood in the context of an extended Heisenberg-Kitaev scenario satisfying all tentative experimental evidence: (i) the maximum of the magnetic susceptibility is located inside the first Brillouin zon…
Motivated by the recent experimental observation of a Mott insulating state for the layered Iridate Na2IrO3, we discuss possible ordering states of the effective Iridium moments in the presence of strong spin-orbit coupling and a magnetic field. For a field pointing in the [111] direction - perpendicular to the hexagonal lattice formed by the Iridium moments - we find that a combination of Heisenberg and Kitaev exchange interactions gives rise to a rich phase diagram with both symmetry breaking magnetically ordered phases as well as a topologically ordered phase that is stable over a small ran…
Kitaev's honeycomb spin-liquid model and its proposed realization in materials such as $α$-RuCl$_3$, Li$_2$IrO$_3$ and Na$_2$IrO$_3$ continue to present open questions about how the dynamics of a spin-liquid are modified in the presence of non-Kitaev interactions as well as the presence of inhomogeneities. Here we use $^{23}$Na nuclear magnetic resonance to probe both static and dynamical magnetic properties in single crystal Na$_2$IrO$_3$. We find that the NMR shift follows the bulk susceptibility above 30 K but deviates from it below; moreover below $T_N$ the spectra show a broad distributio…
Contrary to the original expectation, Na$_2$IrO$_3$ is not a Kitaev's quantum spin liquid (QSL) but shows a zig-zag-type antiferromagnetic order in experiments. Here we propose experimental clues and criteria to measure how a material in hand is close to the Kitaev's QSL state. For this purpose, we systematically study thermal and spin excitations of a generalized Kitaev-Heisenberg model studied by Chaloupka $et$ $al$. in Phys. Rev. Lett. 110, 097204 (2013) and an effective ab initio Hamiltonian for Na$_2$IrO$_3$ proposed by Yamaji $et$ $al$. in Phys. Rev. Lett. 113, 107201 (2014), by employin…
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…
We report inelastic neutron scattering measurements on Na2IrO3, a candidate for the Kitaev spin model on the honeycomb lattice. We observe spin-wave excitations below 5 meV with a dispersion that can be accounted for by including substantial further-neighbor exchanges that stabilize zig-zag magnetic order. The onset of long-range magnetic order below 15.3 K is confirmed via the observation of oscillations in zero-field muon-spin rotation experiments. Combining single-crystal diffraction and density functional calculations we propose a revised crystal structure model with significant departures…
In this work, we investigate the microscopic nature of the magnetism in honeycomb iridium-based systems by performing a systematic study of how the effective magnetic interactions in these compounds depend on various electronic microscopic parameters. We show that the minimal model describing the magnetism in A$_2$IrO$_3$ includes both isotropic and anisotropic Kitaev-type spin-exchange interactions between nearest and next-nearest neighbor Ir ions, and that the magnitude of the Kitaev interaction between next-nearest neighbor Ir magnetic moments is comparable with nearest neighbor interaction…
Iridates are characterized by a fascinating interplay of spin-orbit and electron-electron interactions. The honeycomb-lattice materials A2IrO3 (A=Na,Li) have been proposed to realize pseudospin-1/2 Mott insulating states with strongly anisotropic exchange interactions, described by the Heisenberg-Kitaev model, but other scenarios involving longer-range exchange interactions or more delocalized electrons have been put forward as well. Here we study the influence of non-magnetic doping, i.e., depleted moments, on the magnetic properties of experimentally relevant variants of the Heisenberg-Kitae…
Strong spin-orbital-coupling magnetic systems with the honeycomb structure can provide bond-directional interactions which may result in Kitaev quantum spin liquids and exotic anyonic excitations. However, one of the key ingredients in real materials$-$disorders$-$has been much less studied in Kitaev systems. Here we synthesized a trigonal SrIr$_2$O$_{6-δ}$ with $δ\approx 0.25$, which consists of two-dimensional honeycomb Ir planes with edge-sharing IrO$_6$ octahedra. First-principles computation and experimental measurements suggest that the electronic system is gapped, and there should be no…
We have successfully synthesized the novel antiferromagnets with Ir$^{4+}$ honeycomb lattice ZnIrO$_3$ and MgIrO$_3$ and investigated their magnetic and thermodynamic properties. The two iridates are isomorphic but exhibit qualitatively different magnetic properties. ZnIrO$_3$ shows antiferromagnetic ordering below 46.6 K, whereas MgIrO$_3$ displays weak ferromagnetic behavior below 31.8 K owing to formation of a canted antiferromagnetic ordering. The measurement of magnetic susceptibility with using an oriented powder sample revealed the presence of an XY-like magnetic anisotropy and a tiltin…
Heisenberg interactions are ubiquitous in magnetic materials and have been prevailing in modeling and designing quantum magnets. Bond-directional interactions offer a novel alternative to Heisenberg exchange and provide the building blocks of the Kitaev model, which has a quantum spin liquid (QSL) as its exact ground state. Honeycomb iridates, A2IrO3 (A=Na,Li), offer potential realizations of the Kitaev model, and their reported magnetic behaviors may be interpreted within the Kitaev framework. However, the extent of their relevance to the Kitaev model remains unclear, as evidence for bond-dir…
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…
The electronic ground state in many iridate materials is described by a complex wave-function in which spin and orbital angular momenta are entangled due to relativistic spin-orbit coupling (SOC). Such a localized electronic state carries an effective total angular momentum of $J_{eff}=1/2$. In materials with an edge-sharing octahedral crystal structure, such as the honeycomb iridates Li2IrO3 and Na2IrO3, these $J_{eff}=1/2$ moments are expected to be coupled through a special bond-dependent magnetic interaction, which is a necessary condition for the realization of a Kitaev quantum spin liqui…
Temperature-pressure phase diagram of the Kitaev hyperhoneycomb iridate $β$-Li$_2$IrO$_3$ is explored using magnetization, thermal expansion, magnetostriction, and muon spin rotation ($μ$SR) measurements, as well as single-crystal x-ray diffraction under pressure and ab initio calculations. The Neel temperature of $β$-Li$_2$IrO$_3$ increases with the slope of 0.9 K/GPa upon initial compression, but the reduction in the polarization field $H_c$ reflects a growing instability of the incommensurate order. At 1.4 GPa, the ordered state breaks down upon a first-order transition giving way to a new …
We study the effect of isoelectronic doping and external pressure in tuning the ground state of the honeycomb iridate Na$_2$IrO$_3$ by combining optical spectroscopy with synchrotron x-ray diffraction measurements on single crystals. The obtained optical conductivity of Na$_2$IrO$_3$ is discussed in terms of a Mott insulating picture versus the formation of quasimolecular orbitals and in terms of Kitaev-interactions. With increasing Li content $x$, (Na$_{1-x}$Li$_x$)$_2$IrO$_3$ moves deeper into the Mott insulating regime and there are indications that up to a doping level of 24\% the compound…
We explore the response of Ir $5d$ orbitals to pressure in $β$-$\mathrm{Li_2IrO_3}$, a hyperhoneycomb iridate in proximity to a Kitaev quantum spin liquid (QSL) ground state. X-ray absorption spectroscopy reveals a reconstruction of the electronic ground state below 2 GPa, the same pressure range where x-ray magnetic circular dichroism shows an apparent collapse of magnetic order. The electronic reconstruction, which manifests a reduction in the effective spin-orbit (SO) interaction in $5d$ orbitals, pushes $β$-$\mathrm{Li_2IrO_3}$ further away from the pure $J_{\rm eff}=1/2$ limit. Although l…
Motivated by the experiments on the rare-earth double perovskites, we propose a generalized Kitaev-Heisenberg model to describe the generic interaction between the spin-orbit-entangled Kramers' doublets of the rare-earth moments. We carry out a systematic analysis of the mean-field phase diagram of this new model. In the phase diagram, there exist large regions with a continuous U(1) or O(3) degeneracy. Since no symmetry of the model protects such a continuous degeneracy, we predict that the quantum fluctuation lifts the continuous degeneracy and favors various magnetic orders in the phase dia…
Combining thermodynamic measurements with theoretical density functional and thermodynamic calculations we demonstrate that the honeycomb lattice iridates A2IrO3 (A = Na, Li) are magnetically ordered Mott insulators where the magnetism of the effective spin-orbital S = 1/2 moments can be captured by a Heisenberg-Kitaev (HK) model with Heisenberg interactions beyond nearest-neighbor exchange. Experimentally, we observe an increase of the Curie-Weiss temperature from θ= -125 K for Na2IrO3 to θ= -33 K for Li2IrO3, while the antiferromagnetic ordering temperature remains roughly the same T_N = 15 …
We report results of our study of a newly synthesized honeycomb iridate NaxIrO3 (0.60 < x < 0.80). Single-crystal NaxIrO3 adopts a honeycomb lattice noticeably without distortions and stacking disorder inherently existent in its sister compound Na2IrO3. The oxidation state of the Ir ion is a mixed valence state resulting from a majority Ir5+(5d4) ion and a minority Ir6+(5d3) ion. NaxIrO3 is a Mott insulator likely with a predominant pseudospin = 1 state. It exhibits an effective moment of 1.1 Bohr Magneton/Ir and a Curie-Weiss temperature of -19 K but with no discernable long-range order above…
Recently, there have been contrary claims of Kitaev spin-liquid behaviour and ordered behavior in the honeycomb compound Ag$_3$LiIr$_2$O$_6$ based on various experimental signatures. Our investigations on this system reveal a low-temperature ordered state with persistent dynamics down to the lowest temperatures. Magnetic order is confirmed by clear oscillations in the muon spin relaxation ($μ$SR) time spectrum below 9 K till 52 mK. Coincidentally in $^7$Li nuclear magnetic resonance, a wipe-out of the signal is observed below $\sim$ 10 K which again strongly indicates magnetic order in the low…
Iridate heterostructures are gaining interest as their magnetic properties are much more sensitive to structural distortion compared to pure spin systems due to spin-orbital entanglement induced by strong spin-orbit coupling. While bulk monolayer and bilayer iridates show $ab$-plane canted and $c$-axis antiferromagnetic (AFM) order, recent experiments on layered iridate superlattices (SL) have revealed striking properties, especially in the bilayer SL. A spin model is presented including the tilting induced Kitaev type interactions, which illustrates the proclivity towards $ab$-plane canted AF…
Motivated by the recent synthesis of two insulating Li$_2$IrO$_3$ polymorphs, where Ir$^{4+}$ $S_{eff}$=1/2 moments form 3D ("harmonic") honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $β$-Li$_2$IrO$_3$ hyperhoneycomb lattice and $γ$-Li$_2$IrO$_3$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal 90$^\circ$ Ir-O-Ir bonds, are sizable in these materials. We first consider pure Kitaev interactions, which lead to an exactly soluble 3D quantum spin liquid (QSL) with …
We investigate the dynamical properties of ${\rm Na_2IrO_3}$. For five effective models proposed for ${\rm Na_2IrO_3}$, we numerically calculate dynamical structure factors (DSFs) with an exact diagonalization method. An effective model obtained from $ab$ $initio$ calculations explains inelastic neutron scattering experiments adequately. We further calculate excitation modes based on linearized spin-wave theory. The spin-wave excitation of the effective models obtained by $ab$ $initio$ calculations disagrees with the low-lying excitation of DSFs. We attribute this discrepancy to the location o…
An effective low-energy Hamiltonian of itinerant electrons for iridium oxide Na2IrO3 is derived by an ab initio downfolding scheme. The model is then reduced to an effective spin model on a honeycomb lattice by the strong coupling expansion. Here we show that the ab initio model contains spin-spin anisotropic exchange terms in addition to the extensively studied Kitaev and Heisenberg exchange interactions, and allows to describe the experimentally observed zigzag magnetic order, interpreted as the state stabilized by the antiferromagnetic coupling of the ferromagnetic chains. We clarify possib…
