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arXiv:2303.09156 · Journal of the Physical Society of Japan

Ground-State Phase Diagram of the Kitaev–Heisenberg Model on a Three-dimensional Hyperhoneycomb Lattice

Kiyu Fukui, Yasuyuki Kato, Yukitoshi Motome

Published May 26, 2023• 8 citations
arXivHTMLPDF

Abstract

The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like $β$-Li$_{2}$IrO$_{3}$, $γ$-Li$_2$IrO$_3$, and $β$-ZnIrO$_{3}$ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.

DOI
10.7566/jpsj.92.064708

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The Kitaev model, which hosts a quantum spin liquid (QSL) in the ground state, was originally defined on a two-dimensional honeycomb lattice, but can be straightforwardly extended to any tricoordinate lattices in any spatial dimensions. In particular, the three-dimensional (3D) extensions are of interest as a realization of 3D QSLs, and some materials like βββ-Li2_{2}2​IrO3_{3}3​, γγγ-Li2_22​IrO3_33​, and βββ-ZnIrO3_{3}3​ were proposed for the candidates. However, the phase diagrams of the models for those candidates have not been fully elucidated, mainly due to the limitation of numerical methods for 3D frustrated quantum spin systems. Here we study the Kitaev-Heisenberg model defined on a 3D hyperhoneycomb lattice, by using the pseudofermion functional renormalization group method. We show that the ground-state phase diagram contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases, similar to the two-dimensional honeycomb case. Our results respect the four-sublattice symmetry inherent in the model, which was violated in the previous study. Moreover, we also show how the phase diagram changes with the anisotropy in the interactions. The results provide a reference for the search of the hyperhoneycomb Kitaev materials.
The Kitaev model, whose ground state is a quantum spin liquid (QSL), was originally conceived for spin S=1/2S=1/2S=1/2 moments on a honeycomb lattice. In recent years, the model has been extended to higher SSS from both theoretical and experimental interests, but the stability of the QSL ground state has not been systematically clarified for general SSS, especially in the presence of other additional interactions, which inevitably exist in candidate materials. Here we study the spin-SSS Kitaev-Heisenberg models by using an extension of the pseudofermion functional renormalization group method to general SSS. We show that, similar to the S=1/2S=1/2S=1/2 case, the phase diagram for higher SSS contains the QSL phases in the vicinities of the pristine ferromagnetic and antiferromagnetic Kitaev models, in addition to four magnetically ordered phases. We find, however, that the QSL phases shrink rapidly with increasing SSS, becoming vanishingly narrow for $S\ensuremath{\ge}2$, whereas the phase boundaries between the ordered phases remain almost intact. Our results provide a reference for the search for higher-SSS Kitaev materials.
Motivated by recent experiments on $\ensuremath{\beta}$-Li2{}_{2}2​IrO3{}_{3}3​, we study the phase diagram of the Heisenberg-Kitaev model on a three-dimensional lattice of tricoordinated Ir4+{}^{4+}4+, dubbed the hyperhoneycomb lattice. The lattice geometry of this material, along with Ir4+{}^{4+}4+ ions carrying Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=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 Z2{Z}_{2}Z2​ 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, dc=2{d}_{c}=2dc​=2). We discuss the effect of the magnetic field and finite temperature on different phases that may be relevant for future experiments.
Motivated by the recent synthesis of two insulating Li2IrO3{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}Li2​IrO3​ polymorphs, where Ir4+{\mathrm{Ir}}^{4+}Ir4+ Seff=1/2{S}_{\mathrm{eff}}=1/2Seff​=1/2 moments form 3D (``harmonic'') honeycomb structures with threefold coordination, we study magnetic Hamiltonians on the resulting $\ensuremath{\beta}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ hyperhoneycomb lattice and $\ensuremath{\gamma}\text{\ensuremath{-}}{\mathrm{Li}}_{2}{\mathrm{IrO}}_{3}$ stripyhoneycomb lattice. Experimentally measured magnetic susceptibilities suggest that Kitaev interactions, predicted for the ideal ${90}^{\ensuremath{\circ}}$ 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 Z2{\mathrm{Z}}_{2}Z2​ flux loops. Unlike 2D QSLs, the 3D QSL is stable to finite temperature, with ${T}_{c}\ensuremath{\approx}|K|/100$. On including Heisenberg couplings, exact solubility is lost. However, by noting that the shortest closed loop $\ensuremath{\ell}$ is relatively large in these structures, we construct an $\ensuremath{\ell}\ensuremath{\rightarrow}\ensuremath{\infty}$ 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.
The recently discovered three-dimensional hyperhoneycomb iridate, β -Li 2 IrO 3 , has raised hopes for the realization of the 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, , correctly captures the low-energy electronic structure. The effective spin model derived in the strong-coupling limit supplemented by the ab initio results is shown to be dominated by the Kitaev interaction. We demonstrated that the possible range of parameters is consistent with a non-coplanar spiral magnetic order found in a recent experiment. All of these analyses suggest that β -Li 2 IrO 3 may be the closest among known materials to the Kitaev spin-liquid regime.
Recently there has been considerable excitement surrounding the promising realization of high-spin Kitaev material, such as the quasi-2D compound CrI3_33​ and CrGeTe3_33​. However, the stability of quantum spin liquids (QSL) against single ion anisotropy (SIA) in these materials and the global quantum phase diagram of the extended spin-3/2 Kitaev model with finite SIA remain unclear. In this study, we perform large-scale density matrix renormalization group (DMRG) to explore the quantum phase diagram of the generalized spin-3/2 Kitaev-Heisenberg (K-H) model accompanied with SIA AcA_cAc​. In the Ac=0A_c=0Ac​=0 limit, the spin-3/2 K-H model exhibits a quantum phase diagram similar to that of a spin-1/2 system, including two QSLs around antiferromagnetic and ferromagnetic Kitaev models. For models with finite AcA_cAc​, we map out the quantum phase diagram around two Kitaev points and observe distinct types of in-plane vortex orders developed from these two QSL phases. Interestingly, series of nearly degenerate vortex configurations are discovered in each vortex phases. Using linear spin-wave theory, we demonstrate that these vortex configurations can be understood as a consequence of the quantum correction on a continuous family of degenerate classical states.
Abstract We study the ground state of the 1D Kitaev-Heisenberg (KH) model using the density-matrix renormalization group and Lanczos exact diagonalization methods. We obtain a rich ground-state phase diagram as a function of the ratio between Heisenberg (J = cosϕ) and Kitaev (K = sinϕ) interactions. Depending on the ratio, the system exhibits four long-range ordered states: ferromagnetic-z, ferromagnetic-xy, staggered-xy, Néel-z, and two liquid states: Tomonaga-Luttinger liquid and spiral-xy. The two Kitaev points ϕ=π2{\boldsymbol{\phi }}{\boldsymbol{=}}\frac{{\boldsymbol{\pi }}}{{\bf{2}}}ϕ=2π​ ϕ=π2 and φ=3π2{\boldsymbol{\varphi }}{\boldsymbol{=}}\frac{{\bf{3}}{\boldsymbol{\pi }}}{{\bf{2}}}φ=23π​ φ=3π2 are singular. Theϕ-dependent phase diagram is similar to that for the 2D honeycomb-lattice KH model. Remarkably, all the ordered states of the honeycomb-lattice KH model can be interpreted in terms of the coupled KH chains. We also discuss the magnetic structure of the K-intercalated RuCl3, a potential Kitaev material, in the framework of the 1D KH model. Furthermore, we demonstrate that the low-lying excitations of the 1D KH Hamiltonian can be explained within the combination of the known six-vertex model and spin-wave theory.