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 presence of gapless Goldstone modes. The procedure used to search for the hidden symmetries is quite general and may be extended to other bond-anisotropic spin models and other lattices, such as the triangular, kagome, hyper-honeycomb, or harmonic-honeycomb lattices. We apply our findings to the honeycomb lattice iridates Na2IrO3 and Li2IrO3, and illustrate how they help to identify plausible values of the model parameters that are compatible with the available experimental data.
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…
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…
We investigate the doping effects of magnetic and nonmagnetic impurities injected to the honeycomb iridate sample of Na2IrO3 . Both the doping result in changing the ordering temperature as well as the Curie-Weiss temperature of the parent sample as a consequence of enhancement of the lattice frustration, screening of the Ir atoms and spin-orbit effects that reflects in the susceptibility and specific heat measurements. Our findings are corroborated by a detailed comparative study of various magnetic and nonmagnetic impurity atoms that have notable effects on different electronic properties of…
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 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…
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 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…
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…
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…
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…
The nature of the effective spin Hamiltonian and magnetic order in the honeycomb iridates is explored by considering a trigonal crystal field effect and spin-orbit coupling. Starting from a Hubbard model, an effective spin Hamiltonian is derived in terms of an emergent pseudo-spin-1/2 moment in the limit of large trigonal distortions and spin-orbit coupling. The present pseudo-spins arise from a spin-orbital locking and are different from the jeff = 1/2 moments that are obtained when the spin-orbit coupling dominates and trigonal distortions are neglected. The resulting spin Hamiltonian is ani…
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.
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 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 …
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 $θ_{CW}$ …
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…
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…
We report the successful synthesis of single-crystals of the layered iridate, (Na$_{1-x}$Li$_{x}$)$_2$IrO$_3$, $0\leq x \leq 0.9$, and a thorough study of its structural, magnetic, thermal and transport properties. The new compound allows a controlled interpolation between Na$_2$IrO$_3$ and Li$_2$IrO$_3$, while maintaing the novel quantum magnetism of the honeycomb Ir$^{4+}$ planes. The measured phase diagram demonstrates a dramatic suppression of the Néel temperature, $T_N$, at intermediate $x$ suggesting that the magnetic order in Na$_2$IrO$_3$ and Li$_2$IrO$_3$ are distinct, and that at $x\…
We have successfully synthesized the novel antiferromagnets with Ir$^{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…
The complexity of the antiferromagnetic orders observed in the honeycomb iridates is a double-edged sword in the search for a quantum spin-liquid ground state: both attesting that the magnetic interactions provide many of the necessary ingredients, but simultaneously impeding access. As a result, focus has been drawn to the unusual magnetic orders and the hints they provide to the underlying spin correlations. However, the study of any particular broken symmetry state generally provides little clue as to the possibilities of other nearby ground states \cite{Anderson}. Here we use extreme magne…
The structural and magnetic properties of a new series of iridium-based honeycomb lattices with the formula Na3-delta(Na1-xMgx)Ir2O6 (x = 0 to 1) are reported. As x and delta are increased, the honeycomb lattice contracts and the strength of the antiferromagnetic interactions decreases systematically due to a reduction in Ir-O-Ir bond angles. Samples with imperfect stoichiometry exhibit disordered magnetic freezing at temperatures Tf between 3.4 K and 5 K. This glassy magnetism likely arises due to the presence of non-magnetic Ir3+, which are distributed randomly throughout the lattice, with a…
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…
Direct experimental investigations of the low-energy electronic structure of the Na$_2$IrO$_3$ iridate insulator are sparse and draw two conflicting pictures. One relies on flat bands and a clear gap, the other involves dispersive states approaching the Fermi level, pointing to surface metallicity. Here, by a combination of angle-resolved photoemission, photoemission electron microscopy, and x-ray absorption, we show that the correct picture is more complex and involves an anomalous band, arising from charge transfer from Na atoms to Ir-derived states. Bulk quasiparticles do exist, but in one …
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…
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 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…
Single-crystal x-ray diffraction studies with synchrotron radiation on the honeycomb iridate $α$-Li$_{2}$IrO$_{3}$ reveal a pressure-induced structural phase transition with symmetry lowering from monoclinic to triclinic at a critical pressure of $P_{c}$ = 3.8 GPa. According to the evolution of the lattice parameters with pressure, the transition mainly affects the $ab$ plane and thereby the Ir hexagon network, leading to the formation of Ir--Ir dimers. These observations are independently predicted and corroborated by our \textit{ab initio} density functional theory calculations where we find…
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.…
Two recent theoretical works studied the role of Kitaev interactions in the newly observed incommensurate magnetic order in the hyper-honeycomb ($β$-Li$_2$IrO$_3$) and stripy-honeycomb ($γ$-Li$_2$IrO$_3$) iridates. Each of these works analyzed a different model ($JKΓ$ versus coupled zigzag chain model) using a contrasting method (classical versus soft-spin analysis). The lack of commonality between these works precludes meaningful comparisons and a proper understanding of these unusual orderings. In this study, we complete the unfinished picture initiated by these two works by solving both mod…
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…
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…
We have used resonant inelastic x-ray scattering to reveal optical magnons in a honeycomb lattice iridate $α$-Li$_{2}$IrO$_{3}$. The spectrum in the energy region 20-25 meV exhibits momentum dependence, of which energy is highest at the location of the magnetic Bragg peak, ($\textit{h}, \textit{k}$) = ($\pm$0.32, 0), and lowered toward (0, 0) and ($\pm$1, 0). We compare our data with a linear spin-wave theory based on a generic nearest-neighbor spin model. We find that a dominant bond-directional Kitaev interaction of order 20 meV is required to explain the energy scale observed in our study. …
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…
A burning question in the emerging field of spin-orbit driven insulating iridates, such as Na2IrO3 and Li2IrO3 is whether the observed insulating state should be classified as a Mott-Hubbard insulator derived from a half-filled relativistic j_eff=1/2 band or as a band insulator where the gap is assisted by spin-orbit interaction, or Coulomb correlations, or both. The difference between these two interpretations is that only for the former, strong spin-orbit coupling (lambda >~ W, where W is the band width) is essential. We have synthesized the isostructural and isoelectronic Li2RhO3 and report…
We have synthesized single crystals of Na$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ and polycrystals of Li$_2$(Ir$_{1-x}$Ti$_x$)O$_3$ and studied the effect of magnetic depletion on the magnetic properties by measurements of the magnetic susceptibility, specific heat and magnetocaloric effect at temperatures down to 0.1~K. In both systems, the non-magnetic substitution rapidly changes the magnetically ordered ground state into a spin glass, indicating strong frustration. While for the Li system the Weiss temperature $Θ_{\rm W}$ remains unchanged up to $x=0.55$, a strong decrease $|Θ_{\rm W}|$ is found for th…
We investigated the magnetic properties of the ilmenite-type iridate CdIrO$_3$ with a honeycomb lattice formed by Ir$^{4+}$ ions prepared via a solid-state metathesis. The magnetization measurements with using the powder sample reveal a large effective magnetic moment and a fairly strong antiferromagnetic interaction, indicating a deviation from the Kitaev model. Considering the relationship between magnetism and crystal structure in CdIrO$_3$ with comparing with the other ilmenite-type iridates ZnIrO$_3$ and MgIrO$_3$, we conclude that insulating CdIrO$_3$ cannot be describe as a $J_{\rm eff}…
Strong spin-orbital-coupling magnetic systems with the honeycomb structure can provide bond-directional interactions which may result in Kitaev quantum spin liquids and exotic anyonic excitations. However, one of the key ingredients in real materials$-$disorders$-$has been much less studied in Kitaev systems. Here we synthesized a trigonal SrIr$_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…
Motivated by recent experiments, we consider a generic spin model in the $j_{\text{eff}}=1/2$ basis for the hyperhoneycomb and harmonic-honeycomb iridates. Based on microscopic considerations, the effect of an additional bond-dependent anisotropic spin exchange interaction ($Γ$) beyond the Heisenberg-Kitaev model is investigated. We obtain the magnetic phase diagrams of the hyperhoneycomb and harmonic-honeycomb ($\mathcal{H}\text{--}1$) lattices via a combination of the Luttinger-Tisza approximation, single-$\mathbf{Q}$ variational ansatz, and classical Monte Carlo simulated annealing. The res…
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…
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…
In this work, we investigate whether the Kitaev honeycomb model can serve as a starting point to realize the intriguing physics of the Sachdev-Ye-Kitaev (SYK) model. The starting point is to strain the system, which leads to flat bands reminiscent of Landau levels, thereby quenching the kinetic energy. The presence of weak residual perturbations, such as Heisenberg interactions and the $γ$-term, creates effective interactions between the Majorana modes when projected into the flux-free sector. We assume the resulting interactions to be effectively random. This leads to a bipartite Sachdev-Ye-K…
We study the reduced fidelity and reduced fidelity susceptibility in the Kitaev honeycomb model. It is shown that the reduced fidelity susceptibility of two nearest site manifest itself a peak at the quantum phase transition point, although the one-site reduced fidelity susceptibility vanishes. Our results directly reveal that the reduced fidelity susceptibility can be used to characterize the quantum phase transition in the Kitaev honeycomb model, and thus suggest that the reduced fidelity susceptibility is an accurate marker of the topological phase transition when it is properly chosen, des…
A range of honeycomb-lattice compounds has been proposed and investigated in the search for a topological Kitaev spin liquid. However, sizable Heisenberg interactions and additional symmetry-allowed exchange anisotropies in the magnetic Hamiltonian of these potential Kitaev materials push them away from the pure Kitaev spin-liquid state. Particularly the Kitaev-to-Heisenberg coupling ratio is essential in this respect. With the help of advanced quantum-chemistry methods, we explore how the magnetic coupling ratios depend on pressure in several honeycomb compounds (Na$_2$IrO$_3$, $β$-Li$_2$IrO$…
Emergence of novel quantum ground states in correlated electron systems with strong spin-orbit coupling has been a recent subject of intensive studies. While it has been realized that spin-orbit coupling can provide non-trivial band topology in weakly interacting electron systems, as in topological insulators and semi-metals, the role of electron-electron interaction in strongly spin-orbit coupled systems has not been fully understood. The availability of new materials with significant electron correlation and strong spin-orbit coupling now makes such investigations possible. Many of these mat…
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…
We propose a theoretical model for a gapless spin liquid phase that may have been observed in a recent experiment on $\mathrm{H_3Li Ir_2 O_6}$. Despite the insulating and non-magnetic nature of the material, the specific heat coefficient $C/T \sim 1/\sqrt{T}$ in zero magnetic field and $C/T \sim T/ B^{3/2}$ with finite magnetic field $B$ have been observed. In addition, the NMR relaxation rate shows $1/(T_1T) \sim (C/T)^2$. Motivated by the fact that the interlayer/in-plane lattice parameters are reduced/elongated by the hydrogen-intercalation of the parent compound $\mathrm{Li_2 Ir O_3}$, we …
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 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 study the effects of static magnetic and electric fields on Kitaev's honeycomb model. Using the electric polarization operator appropriate for Kitaev materials, we derive the effective Hamiltonian for the emergent Majorana fermions to second-order in both the electric and magnetic fields. We find that while individually each perturbation does not qualitatively alter Kitaev spin liquid, the cross-term induces a finite chemical potential at each Dirac node, giving rise to a Majorana-Fermi surface. We argue this gapless phase is stable and exhibits typical metallic phenomenology, such as linea…
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 …
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…
We calculate the resonant inelastic X-ray scattering (RIXS) response of the Kitaev honeycomb model, an exactly solvable quantum-spin-liquid model with fractionalized Majorana and flux excitations. We find that the fundamental RIXS channels, the spin-conserving (SC) and the non-spin-conserving (NSC) ones, do not interfere and give completely different responses. SC-RIXS picks up exclusively the Majorana sector with a pronounced momentum dispersion, whereas NSC-RIXS also creates immobile fluxes, thereby rendering the response only weakly momentum dependent, as in the spin structure factor measur…
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…
Low-energy states of quantum spin liquids are thought to involve partons living in a gauge-field background. We study the spectrum of Majorana fermions of Kitaev's honeycomb model on spherical clusters. The gauge field endows the partons with half-integer orbital angular momenta. As a consequence, the multiplicities reflect not the point-group symmetries of the cluster, but rather its projective symmetries, operations combining physical and gauge transformations. The projective symmetry group of the ground state is the double cover of the point group.
