Kitaev-Heisenberg Model Research Articles

Theory of magnetic phase diagrams in hyperhoneycomb and harmonic-honeycomb iridates

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 ($\Gamma$) 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 resulting phase diagrams on both systems show the existence of incommensurate, non-coplanar spiral magnetic orders as well as other commensurate magnetic orders. The spiral orders show counter-propagating spiral patterns, which may be favorably compared to recent experimental results on both iridates. The parameter regime of various magnetic orders and ordering wavevectors are quite similar in both systems. We discuss the implications of our work to recent experiments and also compare our results to those of the two dimensional honeycomb iridate systems.

Density-matrix renormalization group study of the extended Kitaev-Heisenberg model

We study an extended Kitaev-Heisenberg model including additional anisotropic couplings by using two-dimensional density-matrix renormalization group method. Calculating the gound-state energy, entanglement entropy, and spin-spin correlation functions, we make a phase diagram of the extended Kitaev-Heisenberg model around spin-liquid phase. We find a zigzag antiferromagnetic phase, a ferromagnetic phase, a 120-degree antiferromagnetic phase, and two kinds of incommensurate phases around the Kitaev spin-liquid phase. Furthermore, we study the entanglement spectrum of the model and find that entanglement levels in the Kitaev spin-liquid phase are degenerate forming pairs but those in the magnetically ordered phases are non-degenerate. The Schmidt gap defined as the energy difference between the lowest two levels changes at the phase boundary adjacent to the Kitaev spin-liquid phase. However, we find that phase boundaries between magnetically ordered phases do not necessarily agree with the change of the Schmidt gap.

Erratum: Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms [Phys. Rev. B90, 195102 (2014)

Received 28 November 2014DOI:https://doi.org/10.1103/PhysRevB.90.239903©2014 American Physical Society

Three-dimensional quantum spin liquids in models of harmonic-honeycomb iridates and phase diagram in an infinite-Dapproximation

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 $\beta$-Li$_2$IrO$_3$ hyperhoneycomb lattice and $\gamma$-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 emergent Majorana fermions and Z$_2$ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with $T_c \approx |K|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $\ell$ is relatively large in these structures, we construct an $\ell\rightarrow \infty$ approximation by defining the model on the Bethe lattice. The phase diagram of the Kitaev-Heisenberg model on this lattice is obtained directly in the thermodynamic limit, using tensor network states and the infinite-system time-evolving-block-decimation (iTEBD) algorithm. Both magnetically ordered and gapped QSL phases are found, the latter being identified by an entanglement fingerprint.

Magnetism in spin models for depleted honeycomb-lattice iridates: Spin-glass order towards percolation

Iridates are characterized by a fascinating interplay of spin-orbit and electron-electron interactions. The honeycomb-lattice materials ${A}_{2}{\mathrm{IrO}}_{3}$ $(A=\text{Na},\text{Li})$ 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 nonmagnetic doping, i.e., depleted moments, on the magnetic properties of experimentally relevant variants of the Heisenberg-Kitaev and Heisenberg ${J}_{1}\text{\ensuremath{-}}{J}_{2}\text{\ensuremath{-}}{J}_{3}$ models. We generically find that the zigzag order of the clean system is replaced, upon doping, by a spin-glass state with short-ranged zigzag correlations. We determine the spin-glass temperature as a function of the doping level and show that this quantity allows one to assess the importance of longer-range exchange interactions when the doping is driven across the site-percolation threshold of the honeycomb lattice.

Probing the stability of the spin-liquid phases in the Kitaev-Heisenberg model using tensor network algorithms

We study the extent of the spin liquid phases in the Kitaev-Heisenberg model using infinite Projected Entangled-Pair States tensor network ansatz wave functions directly in the thermodynamic limit. To assess the accuracy of the ansatz wave functions we perform benchmarks against exact results for the Kitaev model and find very good agreement for various observables. In the case of the Kitaev-Heisenberg model we confirm the existence of 6 different phases: N\'eel, stripy, ferromagnetic, zigzag and two spin liquid phases. We find finite extents for both spin liquid phases and discontinuous phase transitions connecting them to symmetry-broken phases.

Emergent quantum phases in a frustratedJ1−J2Heisenberg model on the hyperhoneycomb lattice

We investigate possible quantum ground states as well as the classical limit of a frustrated ${J}_{1}\text{\ensuremath{-}}{J}_{2}$ Heisenberg model on the three-dimensional (3D) hyperhoneycomb lattice. Our study is inspired by the recent discovery of $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$, where ${\mathrm{Ir}}^{4+}$ ions form a 3D network with each lattice site being connected to three nearest neighbors. We focus on the influence of magnetic frustration caused by the second-nearest-neighbor spin interactions. Such interactions are likely to be significant due to the large extent of $5d$ orbitals in iridates or other $5d$ transition metal oxides. In the classical limit, the ground state manifold is given by line degeneracies of the spiral magnetic-order wave vectors when ${J}_{2}/{J}_{1}\ensuremath{\gtrsim}0.17$ while the collinear stripy order is included in the degenerate manifold when ${J}_{2}/{J}_{1}=0.5$. Quantum order-by-disorder effects are studied using both the semiclassical $1/S$ expansion in the spin-wave theory and the Schwinger-boson approach. In general, certain coplanar spiral orders are chosen from the classical degenerate manifold for a large fraction of the phase diagram. Nonetheless quantum fluctuations favor the collinear stripy order over the spiral orders in an extended parameter region around ${J}_{2}/{J}_{1}=0.5$, despite the spin-rotation invariance of the underlying Hamiltonian. This is in contrast to the emergence of stripy order in the Heisenberg-Kitaev model studied earlier on the same lattice, where the Kitaev-type Ising interactions are important for stabilizing the stripy order. As quantum fluctuations become stronger, $U(1)$ and ${Z}_{2}$ quantum spin liquid phases are shown to arise via quantum disordering of the N\'eel, stripy, and spiral magnetically ordered phases. The effects of magnetic anisotropy and their relevance to future experiments are also discussed.

Raman scattering signatures of Kitaev spin liquids in A(2)IrO(3) iridates with A=Na or Li.

We show how Raman spectroscopy can serve as a valuable tool for diagnosing quantum spin liquids (QSL). We find that the Raman response of the gapless QSL of the Kitaev-Heisenberg model exhibits signatures of spin fractionalization into Majorana fermions, which give rise to a broad signal reflecting their density of states, and Z(2) gauge fluxes, which also contribute a sharp feature. We discuss the current experimental situation and explore more generally the effect of breaking the integrability on response functions of Kitaev spin liquids.

Frustration and Entanglement in Compass and Spin-Orbital Models

We review the consequences of intrinsic frustration of the orbital superexchange and of spin-orbital entanglement. While Heisenberg perturbing interactions remove frustration in the compass model, the lowest columnar excitations are robust in the nanoscopic compass clusters and might be used for quantum computations. Entangled spin-orbital states determine the ground states in some cases, while in others concern excited states and lead to measurable consequences, as in the $R$VO$_3$ perovskites. On-site entanglement for strong spin-orbit coupling generates the frustrated Kitaev-Heisenberg model with a rich magnetic phase diagram on the honeycomb lattice. Frustration is here reflected in hole propagation which changes from coherent in an antiferromagnet via hidden quasiparticles in zigzag and stripe phases to entirely incoherent one in the Kitaev spin liquid.

Spiral order in the honeycomb iridateLi2IrO3

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 zone, (ii) the Curie-Weiss temperature is negative relating to dominant antiferromagnetic fluctuations, and (iii) significant second-neighbor spin-exchange is involved.

Zeeman-field-induced topological phase transitions in triplet superconductors

We develop a general Ginzburg-Landau theory which describes the effect of a Zeeman field on the superconducting order parameter in triplet superconductors. Starting from Ginzburg-Landau theories that describe fully gapped time-reversal symmetric triplet superconductors, we show that the Zeeman field has dramatic effects on the topological properties of the superconductors. In particular, in the vicinity of a critical chemical potential separating two topologically distinct phases, it is possible to induce a phase transition to a topologically nontrivial phase which supports chiral edge modes. Moreover, for specific directions of the Zeeman field, we obtain nodal superconducting phases with an emerging chiral symmetry, and with Majorana flat bands at the edge. The Ginzburg-Landau theory is microscopically supported by a self-consistent mean-field theory of the doped Kitaev-Heisenberg model.

Unconventional pairing and electronic dimerization instabilities in the doped Kitaev-Heisenberg model

We study the quantum many-body instabilities of the t−JK−JH Kitaev-Heisenberg Hamiltonian on the honeycomb lattice as a minimal model for a doped spin-orbit Mott insulator. This spin-1/2 model is believed to describe the magnetic properties of the layered transition-metal oxide Na2IrO3. We determine the ground state of the system with finite charge-carrier density from the functional renormalization group (fRG) for correlated fermionic systems. To this end, we derive fRG flow equations adapted to the lack of full spin-rotational invariance in the fermionic interactions, here represented by the highly frustrated and anisotropic Kitaev exchange term. Additionally employing a set of the Ward identities for the Kitaev-Heisenberg model, the numerical solution of the flow equations suggests a rich phase diagram emerging upon doping charge carriers into the ground-state manifold (Z2 quantum spin liquids and magnetically ordered phases). We corroborate superconducting triplet p-wave instabilities driven by ferromagnetic exchange and various singlet pairing phases. For filling δ>1/4, the p-wave pairing gives rise to a topological state with protected Majorana edge modes. For antiferromagnetic Kitaev and ferromagnetic Heisenberg exchanges, we obtain bond-order instabilities at van Hove filling supported by nesting and density-of-states enhancement, yielding dimerization patterns of the electronic degrees of freedom on the honeycomb lattice. Further, our flow equations are applicable to a wider class of model Hamiltonians.11 MoreReceived 1 May 2014Revised 30 June 2014DOI:https://doi.org/10.1103/PhysRevB.90.045135©2014 American Physical Society

Order-by-disorder and spin-orbital liquids in a distorted Heisenberg-Kitaev model

The microscopic modeling of spin-orbit entangled $j=1/2$ Mott insulators such as the layered hexagonal Iridates Na$_2$IrO$_3$ and Li$_2$IrO$_3$ has spurred an interest in the physics of Heisenberg-Kitaev models. Here we explore the effect of lattice distortions on the formation of the collective spin-orbital states which include not only conventionally ordered phases but also gapped and gapless spin-orbital liquids. In particular, we demonstrate that in the presence of spatial anisotropies of the exchange couplings conventionally ordered states are formed through an order-by-disorder selection which is not only sensitive to the type of exchange anisotropy but also to the relative strength of the Heisenberg and Kitaev couplings. The spin-orbital liquid phases of the Kitaev limit -- a gapless phase in the vicinity of spatially isotropic couplings and a gapped Z$_2$ phase for a dominant spatial anisotropy of the exchange couplings -- show vastly different sensitivities to the inclusion of a Heisenberg exchange. While the gapless phase is remarkably stable, the gapped Z$_2$ phase quickly breaks down in what might be a rather unconventional phase transition driven by the simultaneous condensation of its elementary excitations.

Hole propagation in the Kitaev-Heisenberg model: From quasiparticles in quantum Néel states to non-Fermi liquid in the Kitaev phase

We explore with exact diagonalization the propagation of a single hole in four magnetic phases of the $t\text{\ensuremath{-}}J$-like Kitaev-Heisenberg model on a honeycomb lattice: the N\'eel antiferromagnetic, stripe, zigzag, and Kitaev spin-liquid phases. We find coherent propagation of spin-polaron quasiparticles in the antiferromagnetic phase by a mechanism similar to that in the $t\text{\ensuremath{-}}J$ model for high-${T}_{c}$ cuprates. In the stripe and zigzag phases clear quasiparticles features appear in spectral functions of those propagators where holes are created and annihilated on one sublattice, while they remain largely hidden in those spectral functions that correspond to photoemission experiments. As the most surprising result, we find a totally incoherent spectral weight distribution for the spectral function of a hole moving in the Kitaev spin-liquid phase in the strong-coupling regime $t\ensuremath{\gg}J$ relevant for iridates. At intermediate coupling the finite systems calculation reveals a well-defined quasiparticle at the $\ensuremath{\Gamma}$ point; however, we find that the gapless spin excitations wipe out quasiparticles at finite momenta. Also for this more subtle case we conclude that in the thermodynamic limit the lightly doped Kitaev liquid phase does not support quasiparticle states in the neighborhood of $\ensuremath{\Gamma}$, and therefore yields a non-Fermi liquid, contrary to earlier suggestions based on slave-boson studies. These observations are supported by the presented study of the dynamic spin-structure factor in the Kitaev spin liquid regime.

Spin-orbit coupling in octamers in the spinel sulfideCuIr2S4: Competition between spin-singlet and quadrupolar states and its relevance to remnant paramagnetism

We theoretically investigate magnetic properties in the low-temperature phase with the formation of eight-site clusters, octamers, in the spinel compound CuIr$_2$S$_4$. The octamer state was considered to be a spin-singlet state induced by a Peieirls instability through the strong anisotropy of $d$ orbitals, the so-called orbital Peierls state. We reexamine this picture by taking into account the spin-orbit coupling which was ignored in the previous study. We derive a low-energy effective model between $j_{\rm eff}=1/2$ quasispins on Ir$^{4+}$ cations in an octamer from the multiorbital Hubbard model with the strong spin-orbit coupling by performing the perturbation expansion from the strong correlation limit. The effective Hamiltonian is in the form of the Kitaev-Heisenberg model but with an additional interaction, a symmetric off-diagonal exchange interaction originating from the perturbation process including both d-d and d-p-d hopping. Analyzing the effective Hamiltonian on two sites and the octamer by the exact diagonalization, we find that there is a competition between a spin-singlet state and a quadrupolar state. The former singlet state is a conventional one, adiabatically connected to the orbital-Peierls state. On the other hand, the latter quadrupolar state is stabilized by the additional interaction, which consists of a linear combination of different total spin momenta along the spin quantization axis. In the competing region, the model exhibits paramagnetic behavior with a renormalized small effective moment at low temperature. This peculiar remnant paramagnetism is not obtained in the Kitaev-Heisenberg model without the additional interaction. Our results renew the picture of the octamer state and provide a scenario for the intrinsic paramagnetic behavior recently observed in a muon spin rotation experiment [K. M. Kojima et al., Phys. Rev. Lett. 112, 087203 (2014)].

The expanded triangular Kitaev–Heisenberg model in the full parameter space

Abstract The classical Kitaev–Heisenberg model on the triangular lattice is investigated by simulation in its full parameter space together with the next-nearest neighboring Heisenberg interaction or the single-ion anisotropy. The variation of the system is demonstrated directly by the joint density of states (DOS) depending on energy and magnetization obtained from Wang–Landau algorithm. The Metropolis Monte Carlo simulation and the zero-temperature Glauber dynamics are performed to show the internal energy, the correlation functions and spin configurations at zero temperature. It is revealed that two types of DOS (U and inverse U) divide the whole parameter range into two main parts with antiferromagnetic and ferromagnetic features respectively. In the parameter range of U type DOS, the mixed frustration from the triangular geometry and the Kitaev interaction produces rich phases, which are influenced in different ways by the next-nearest neighboring Heisenberg interaction and the single-ion anisotropy.

Structural, magnetic, and electrical properties of Li2Ir1−xRuxO3

The crystal structure, resistivity, and magnetic susceptibility of the Li${}_{2}$Ir${}_{1\ensuremath{-}x}$Ru${}_{x}$O${}_{3}$ ($x=0$--1) polycrystals have been investigated. We found that the parent antiferromagnetic phase disappears for $x>0.2$ and bond dimers appear in the averaged structure for $x>0.5$ and likely fluctuate for much smaller $x$. Unexpectedly, this system remains insulating for all the doping levels, contrary to the predictions based on the one-band ${j}_{\mathrm{eff}}=1/2$ Kitaev-Heisenberg model. These results suggest that the honeycomb iridates doped with ruthenium are a unique $5d$-orbital-based platform for studying the interplay of the charge, orbital, spin, and lattice degrees of freedom.

Order-by-disorder and magnetic field response in the Heisenberg-Kitaev model on a hyperhoneycomb lattice

We study the finite temperature phase diagram of the Heisenberg-Kitaev model on a three dimensional hyperhoneycomb lattice. Using semiclassical analysis and classical Monte-Carlo simulations, we investigate quantum and thermal order-by-disorder, as well as the magnetic ordering temperature. We find the parameter regime where quantum and thermal fluctuations favor different magnetic orders, which leads to an additional finite temperature phase transition within the ordered phase. This transition, however, occurs at a relatively low temperature and the entropic effects may dominate most of the finite temperature region below the ordering temperature. In addition, we explore the magnetization process in the presence of a magnetic field and discover spin-flop transitions which are sensitive to the applied-field direction. We discuss implications of our results for future experiments on a hyperhoneycomb lattice system such as the recently discovered \beta-Li2IrO3.

Kitaev-Heisenberg models for iridates on the triangular, hyperkagome, kagome, fcc, and pyrochlore lattices

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 Ir${}_{2}$O${}_{4}$ and Na${}_{4}$Ir${}_{3}$O${}_{8}$, 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 of these lattices and even to certain additional further neighbor Heisenberg couplings. A Klein four-group $\ensuremath{\cong}$${\mathbb{Z}}_{2}\ifmmode\times\else\texttimes\fi{}{\mathbb{Z}}_{2}$ structure is associated with this mapping (hence Klein duality). A finite lattice admits the duality if a simple geometrical condition is met. This duality predicts fluctuation-free ordered states on these different 2D and 3D lattices, which are analogues of the honeycomb lattice KH stripy order. This result is used in conjunction with a semiclassical Luttinger-Tisza approximation to obtain phase diagrams for KH models on the different lattices. We also discuss a Majorana fermion based mean-field theory at the Kitaev point, which is exact on the honeycomb lattice, for the KH models on the different lattices. We attribute the rich behavior of these models to the interplay of geometric frustration and frustration induced by spin-orbit coupling.

Heisenberg-Kitaev model on the hyperhoneycomb lattice

Motivated by recent experiments on $\ensuremath{\beta}$-Li${}_{2}$IrO${}_{3}$, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir${}^{4+}$, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir${}^{4+}$ ions carrying ${J}_{\mathrm{eff}}=1/2$ moments, suggests that the Heisenberg-Kitaev model may effectively capture the low-energy spin-physics of the system in the strong-coupling limit. Using a combination of semiclassical analysis, exact solution, and slave-fermion mean-field theory, we find, in addition to the spin liquid, four different magnetically ordered phases depending on the parameter regime. All four magnetic phases---the N\'eel, the polarized ferromagnet, the skew-stripy, and the skew-zig-zag---have collinear spin ordering. The three-dimensional ${Z}_{2}$ spin liquid, which extends over an extended parameter regime around the exactly solvable Kitaev point, has a gapless Majorana mode with a deformed Fermi circle (codimensions, ${d}_{c}=2$). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.