Kitaev-Heisenberg Model Research Articles

Bond disorder in extended Heisenberg-Kitaev models: Spin textures and in-gap states in the high-field regime

Bond disorder in extended Heisenberg-Kitaev models: Spin textures and in-gap states in the high-field regime

Cobalt-based pyroxenes: A new playground for Kitaev physics

Recent advances in the study of cobaltites have unveiled their potential as a promising platform for realizing Kitaev physics in honeycomb systems and the Ising model in weakly coupled chain materials. In this manuscript, we explore the magnetic properties of pyroxene SrCoGe[Formula: see text]O[Formula: see text] using a combination of neutron scattering, ab initio methods, and linear spin-wave theory. Through careful examination of inelastic neutron scattering powder spectra, we propose a modified Kitaev model to accurately describe the twisted chains of edge-sharing octahedra surrounding Co[Formula: see text] ions. The extended Kitaev-Heisenberg model, including a significant anisotropic bond-dependent exchange term with [Formula: see text], is identified as the key descriptor of the magnetic interactions in SrCoGe[Formula: see text]O[Formula: see text]. Furthermore, our heat capacity measurements reveal an effect of an external magnetic field (approximately 13 T) which shifts the system from a fragile antiferromagnetic ordering with [Formula: see text] K to a field-induced state. We argue that pyroxenes, particularly those modified by substituting Ge with Si and its less extended [Formula: see text] orbitals, emerge as a platform for the Kitaev model. This opens up possibilities for advancing our understanding of Kitaev physics.

Spin- S Kitaev-Heisenberg model on the honeycomb lattice: A high-order treatment via the many-body coupled cluster method

We study the spin-S Kitaev-Heisenberg model on the honeycomb lattice for S=1/2, 1, and 3/2, by using the coupled cluster method (CCM) of microscopic quantum many-body theory. This system is one of the earliest extensions of the Kitaev model and is believed to contain two extended spin liquid phases for any value of the spin quantum number S. We show that the CCM delivers accurate estimates for the phase boundaries of these spin liquid phases, as well as other transition points in the phase diagram. Moreover, we find evidence of two unexpected narrow phases for S=1/2, one sandwiched between the zigzag and ferromagnetic phases and the other between the Néel and the stripy phases. The results establish the CCM as a versatile numerical technique that can capture the strong quantum-mechanical fluctuations that are inherently present in generalized Kitaev models with competing bond-dependent anisotropies. Published by the American Physical Society 2024

Emergence of vortex state in the S=1 Kitaev-Heisenberg model with single-ion anisotropy

The search for Kitaev spin liquid states has recently broadened to include a number of honeycomb materials with integer spin moments. The qualitative difference with their spin-1/2 counterparts is the presence of single-ion anisotropy (SIA). This motivates our investigation of the effects of SIA on the ground state of the spin-1 Kitaev-Heisenberg (KH) model using the density-matrix renormalization group which allows construction of detailed phase diagrams around the Kitaev points. We demonstrate that positive out-of-plane SIA induces an in-plane vortex state without the need for off-diagonal interactions. Conversely, negative SIA facilitates the emergence of a state in the presence of antiferromagnetic Heisenberg interactions, whereas a state can emerge for ferromagnetic Heisenberg coupling. These findings, pertinent even for weak SIA, not only enhance our theoretical understanding of the spin-1 KH model but also suggest experimental prospects for observing these novel magnetic states in material realizations. Published by the American Physical Society 2024

Kitaev-Heisenberg model on the star lattice: From chiral Majorana fermions to chiral triplons

The interplay of frustrated interactions and lattice geometry can lead to a variety of exotic quantum phases. Here we unearth a particularly rich phase diagram of the Kitaev-Heisenberg model on the star lattice, a triangle decorated honeycomb lattice breaking sublattice symmetry. In the antiferromagnetic regime, the interplay of Heisenberg coupling and geometric frustration leads to the formation of valence bond solid (VBS) phases—a singlet VBS and a bond selective triplet VBS stabilized by the Kitaev exchange. We show that the ratio of the Kitaev versus Heisenberg exchange tunes between these VBS phases and chiral quantum spin-liquid regimes. Remarkably, the VBS phases host a whole variety of chiral triplon excitations with high Chern numbers in the presence of a weak magnetic field. We discuss our results in light of a recently synthesized star lattice material and other decorated lattice systems. Published by the American Physical Society 2024

Electric-field control of magnetic anisotropies: Applications to Kitaev spin liquids and topological spin textures

Magnetic anisotropies often originate from the spin-orbit coupling and determine magnetic ordering patterns. We develop a microscopic theory for dc electric-field controls of magnetic anisotropies in magnetic Mott insulators and discuss its applications to Kitaev materials and topological spin textures. Throughout this paper, we take a microscopic approach based on Hubbard-type lattice models, tight-binding models with onsite interactions. We derive a low-energy spin Hamiltonian from a fourth-order perturbation expansion of the Hubbard-type model. We show in the presence of a strong intra-atomic spin-orbit coupling that dc electric fields add non-Kitaev interactions such as a Dzyaloshinskii-Moriya interaction and an off-diagonal Γ′ interaction to the Kitaev-Heisenberg model and can induce a topological quantum phase transition between Majorana Chern insulating phases. We also investigate the interatomic Rashba spin-orbit coupling and its effects on topological spin textures. dc electric fields turn out to create and annihilate magnetic skyrmions, hedgehogs, and chiral solitons. We propose several methods of creating topological spin textures with external electromagnetic fields. Our theory clarifies that the strong but feasible electric field can control Kitaev spin liquids and topological spin textures. Published by the American Physical Society 2024

Machine learning reveals features of spinon Fermi surface

With rapid progress in simulation of strongly interacting quantum Hamiltonians, the challenge in characterizing unknown phases becomes a bottleneck for scientific progress. We demonstrate that a Quantum-Classical hybrid approach (QuCl) of mining sampled projective snapshots with interpretable classical machine learning can unveil signatures of seemingly featureless quantum states. The Kitaev-Heisenberg model on a honeycomb lattice under external magnetic field presents an ideal system to test QuCl, where simulations have found an intermediate gapless phase (IGP) sandwiched between known phases, launching a debate over its elusive nature. We use the correlator convolutional neural network, trained on labeled projective snapshots, in conjunction with regularization path analysis to identify signatures of phases. We show that QuCl reproduces known features of established phases. Significantly, we also identify a signature of the IGP in the spin channel perpendicular to the field direction, which we interpret as a signature of Friedel oscillations of gapless spinons forming a Fermi surface. Our predictions can guide future experimental searches for spin liquids.

Majorana-fermion mean field theories of Kitaev quantum spin liquids

We determine the phase diagrams of anisotropic Kitaev-Heisenberg models on the honeycomb lattice using parton mean-field theories based on different Majorana fermion representations of the S=1/2 spin operators. First, we use a two-dimensional Jordan-Wigner transformation (JWT) involving a semi-infinite snake string operator. To ensure that the fermionized Hamiltonian remains local, we consider the limit of extreme Ising exchange anisotropy in the Heisenberg sector. Second, we use the conventional Kitaev representation in terms of four Majorana fermions subject to local constraints, which we enforce through Lagrange multipliers. For both representations, we self-consistently decouple the interaction terms in the bond and magnetization channels and determine the phase diagrams as a function of the anisotropy of the Kitaev couplings and the relative strength of the Ising exchange. While both mean-field theories produce identical phase boundaries for the topological phase transition between the gapless and gapped Kitaev quantum spin liquids, the JWT fails to correctly describe the magnetic instability and finite-temperature behavior. Our results show that the magnetic phase transition is first order at low temperatures but becomes continuous above a certain temperature. At this energy scale, we also observe a finite-temperature crossover on the quantum-spin-liquid side, from a fractionalized paramagnet at low temperatures, in which gapped flux excitations are frozen out, to a conventional paramagnet at high temperatures. Published by the American Physical Society 2024

Static and dynamical magnetic properties of the extended Kitaev-Heisenberg model with spin vacancies

Static and dynamical magnetic properties of the extended Kitaev-Heisenberg model with spin vacancies

Thermal Hall conductivity near field-suppressed magnetic order in a Kitaev-Heisenberg model

Historically, analytical treatments for the thermal Hall effect of many-body systems have been based around an underlying quasiparticle assumption. The authors introduce here a new purification-based tensor network method for the thermal Hall response of gapped systems, making no prior quasiparticle assumption. Motivated by reports of half-quantized thermal Hall response of Kitaev materials near field-suppressed magnetic order, the authors calculate the thermal Hall response in this parameter regime and find that it is large, but non-universal, ruling out the possibility of a revival of Ising topological order and a Majorana Hall state through this route.

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

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 $\beta$-Li$_{2}$IrO$_{3}$, $\gamma$-Li$_2$IrO$_3$, and $\beta$-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.

Electronic transport mechanisms in a thin crystal of the Kitaev candidate α -RuCl3 probed through guarded high impedance measurements

α-RuCl3 is considered to be the top candidate material for the experimental realization of the celebrated Kitaev model, where ground states are quantum spin liquids with interesting fractionalized excitations. It is, however, known that additional interactions beyond the Kitaev model trigger in α-RuCl3 a long-range zigzag antiferromagnetic ground state. In this work, we investigate a nanoflake of α-RuCl3 through guarded high impedance measurements aimed at reaching the regime where the system turns into a zigzag antiferromagnet. We investigated a variety of temperatures (1.45–175 K) and out-of-plane magnetic fields (up to 11 T), finding a clear signature of a structural phase transition at ≈160 K as reported for thin crystals of α-RuCl3, as well as a thermally activated behavior at temperatures above ≈30 K, with a characteristic activation energy significantly smaller than the energy gap that we observe for α-RuCl3 bulk crystals through our angle resolved photoemission spectroscopy (ARPES) experiments. Additionally, we found that below ≈30 K, transport is ruled by Efros–Shklovskii variable range hopping (VRH). Most importantly, our data show that below the magnetic ordering transition known for bulk α-RuCl3 in the frame of the Kitaev–Heisenberg model (≈7 K), there is a clear deviation from VRH or thermal activation transport mechanisms. Our work demonstrates the possibility of reaching, through specialized high impedance measurements, the thrilling ground states predicted for α-RuCl3 at low temperatures in the frame of the Kitaev–Heisenberg model and informs about the transport mechanisms in this material in a wide temperature range.

Sonication-assisted liquid exfoliation and size-dependent properties of magnetic two-dimensional α-RuCl3

Originating from the hexagonal arrangement of magnetic ions in the presence of strong spin orbit coupling, α-RuCl3 is considered as model system for the Kitaev-Heisenberg model. While the magnetic properties of α-RuCl3 have been studied in bulk single crystals or micromechanically-exfoliated nanosheets, little is known about the nanosheets’ properties after exfoliation by techniques suitable for mass production such as liquid phase exfoliation (LPE). Here, we demonstrate sonication-assisted LPE on α-RuCl3 single crystals in an inert atmosphere. Coupled with centrifugation-based size selection techniques, the accessible size- and thickness range is quantified by statistical atomic force microscopy. Individual nanosheets obtained after centrifugation-based size selection are subjected to transmission electron microscopy to confirm their structural integrity after the exfoliation. The results are combined with bulk characterisation methods, including Raman and x-ray photoelectron spectroscopy, and powder diffraction experiments to evaluate the structural integrity of the nanosheets. We report changes of the magnetic properties of the nanomaterial with nanosheet size, as well as photospectroscopic metrics for the material concentration and average layer number. Finally, a quantitative analysis on environmental effects on the nanomaterial integrity is performed based on time and temperature dependent absorbance spectroscopy revealing a relatively slow decay (half-life of ∼2000 h at 20 °C), albeit with low activation energies of 6–20 kJ mol−1.

Disorder effects in the Kitaev-Heisenberg model

We study the interplay of disorder and Heisenberg interactions in the Kitaev model on a honeycomb lattice. The effect of disorder on the transition between Kitaev spin liquid and magnetic ordered states as well as the stability of magnetic ordering is investigated. Using Lanczos exact diagonalization we discuss the consequences of two types of disorder: (i) random-coupling disorder and (ii) singular-coupling disorder. They exhibit qualitatively similar effects in the pure Kitaev-Heisenberg model without long-range interactions. The range of spin-liquid phases is reduced and the transition to magnetic ordered phases becomes more crossoverlike. Furthermore, the long-range zigzag and stripy orderings in the clean system are replaced by their three domains with different ordering direction. Especially in the crossover range the coexistence of magnetically ordered and Kitaev spin-liquid domains is possible. With increasing the disorder strength the area of domains becomes smaller and the system goes into a spin-glass state. However, the disorder effect is different in magnetically ordered phases caused by long-range interactions. The stability of such magnetic ordering is diminished by singular-coupling disorder and, accordingly, the range of the spin-liquid regime is extended. This mechanism may be relevant to materials like $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ and ${\mathrm{H}}_{3}{\mathrm{LiIr}}_{2}{\mathrm{O}}_{6}$ where the zigzag ground state is stabilized by weak long-range interactions. We also find that the flux gap closes at a critical disorder strength and vortices appears in the flux arrangement. Interestingly, the vortices tend to form kinds of commensurate ordering.

Thermal Hall conductivity with sign change in the Heisenberg–Kitaev kagome magnet

The Heisenberg–Kitaev (HK) model on various lattices has attracted a lot of attention because it may lead to exotic states such as quantum spin liquid and topological orders. The rare-earth-based kagome lattice (KL) compounds and have q = 0, 120° order and canted ferromagnetic (CFM) order, respectively. Interestingly, the HK model on the KL has the same ground state long-range orders. In the theoretical phase diagram, the CFM phase resides in a continuous parameter region and there is no phase change across special parameter points, such as the Kitaev ferromagnetic (KFM) point, the ferromagnetic (FM) point and its dual FM point. However, a ground state property cannot distinguish a system with or without topological nontrivial excitations and related phase transitions. Here, we study the topological magnon excitations and related thermal Hall conductivity in the HK model on the KL with CFM order. The CFM phase can be divided into two regions related by the Klein duality, with the self dual KFM point as their boundary. We find that the scalar spin chirality, which is intrinsic in the CFM order, changes sign across the KFM point. This leads to the opposite Chern numbers of corresponding magnon bands in the two regions, and also the sign change of the magnon thermal Hall conductivity.

Easy-plane anisotropic-exchange magnets on a honeycomb lattice: Quantum effects and dealing with them

We provide analytical and numerical insights into the phase diagram and other properties of the extended Kitaev-Heisenberg model on the honeycomb lattice in the {\it easy-plane} limit, in which interactions are only between spin components that belong to the plane of magnetic ions. This parameter subspace allows for a much-needed systematic {\it quantitative} investigation of spin excitations in the ordered phases and of their generic features. Specifically, we demonstrate that in this limit one can consistently take into account magnon interactions in both zero-field zigzag and field-polarized phases. For the nominally polarized phase, we propose a regularization of the unphysical divergences that occur at the critical field and are plaguing the $1/S$-approximation in this class of models. For the explored parameter subspace, all symmetry-allowed terms of the standard parametrization of the extended Kitaev-Heisenberg model, such as $K$, $J$, and $\Gamma$, are significant, making the offered consideration relevant to a much wider parameter space. The dynamical structure factor near paramagnetic critical point illustrates this relevance by showing features that are reminiscent of the ones observed in $\alpha$-RuCl$_3$, underscoring that they are not unique and should be common to a wide range of parameters of the model and, by an extension, to other materials.

Ground-state phase diagram of spin- S Kitaev-Heisenberg models

The Kitaev model, whose ground state is a quantum spin liquid (QSL), was originally conceived for spin $S=1/2$ moments on a honeycomb lattice. In recent years, the model has been extended to higher $S$ from both theoretical and experimental interests, but the stability of the QSL ground state has not been systematically clarified for general $S$, especially in the presence of other additional interactions, which inevitably exist in candidate materials. Here we study the spin-$S$ Kitaev-Heisenberg models by using an extension of the pseudofermion functional renormalization group method to general $S$. We show that, similar to the $S=1/2$ case, the phase diagram for higher $S$ 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 $S$, becoming vanishingly narrow for $S\geq 2$, whereas the phase boundaries between the ordered phases remain almost intact. Our results provide a reference for the search of higher-$S$ Kitaev materials.

Negative sign free formulations of generalized Kitaev models with higher symmetries

We provide a negative-sign-free formulation of the auxiliary field quantum Monte Carlo algorithm for generalized Kitaev models with higher symmetries. Our formulation is based on the Abrikosov fermion representation of the spin-1/2 degree of freedom and the phase pinning approach [Phys. Rev. B 104, L081106 (2021)]. Enhancing the number of fermion flavors or orbitals from one to $N$ allows one to generalize the inherent $Z_2$ global symmetry to Z$_2$$\times$SU($N$)$_o$. Using this general approach, we study the Z$_2$$\times$SU(2)$_o$ Kitaev-Heisenberg model reflecting the competition between the isotropic Heisenberg exchange and Kitaev-type bond-directional exchange interactions. We show that the symmetry enhancement provides a path to escape frustration and that the spin liquid phases in the original Z$_2$ symmetric model are not present in this model. Nevertheless, the ground-state phase diagram is extremely rich and has points with higher global and local continuous symmetries as well as de-confined quantum critical points.

Stability of ordered and disordered phases in the Heisenberg-Kitaev model in a magnetic field

The S = 1/2 Kitaev honeycomb model has attracted significant attention as an exactly solvable example with a quantum spin liquid ground state. In an properly oriented external magnetic field, the system exhibits chiral Majorana edge modes with an associated quantized thermal Hall conductance, and a distinct spin-disordered phase emerges at intermediate field strengths, below the polarized phase. However, since material realizations of Kitaev magnetism invariably display competing exchange interactions, the stability of these exotic phases with respect to additional couplings is a key issue. Here, we report a 24-site exact diagonalization study of the Heisenberg-Kitaev model in a magnetic field applied in the [001] and [111] directions. By mapping the full phase diagram of the model and contrasting the results to recent nonlinear spin-wave calculations, we show that both methods agree well, thus establishing that quantum corrections substantially modify the classical phase diagram. Furthermore, we find that, in a [111] field, the intermediate-field spin-disordered phase is remarkably stable to Heisenberg interactions and may potentially end in a novel quantum tricritical point.

Quantum many-body scars in spin-1 Kitaev chains

To provide a physical example of quantum scars, we study the many-body scars in the spin-1 Kitaev chain where the so-called PXP Hamiltonian is exactly embedded in the spectra. Regarding the conserved quantities, the Hilbert space is fragmented into disconnected subspaces and we explore the associated constrained dynamics. The continuous revivals of the fidelity and the entanglement entropy when the initial state is prepared in $\vert\mathbb{Z}_k\rangle$ ($k=2,3$) state illustrate the essential physics of the PXP model. We study the quantum phase transitions in the one-dimensional spin-1 Kitaev-Heisenberg model using the density-matrix renormalization group and Lanczos exact diagonalization methods, and determine the phase diagram. We parametrize the two terms in the Hamiltonian by the angle $\phi$, where the Kitaev term is $K\equiv\sin(\phi)$ and competes with the Heisenberg $J\equiv\cos(\phi)$ term. One finds a rich ground state phase diagram as a function of the angle $\phi$. Depending on the ratio $K/J\equiv\tan(\phi)$, the system either breaks the symmetry to one of distinct symmetry broken phases, or preserves the symmetry in a quantum spin liquid phase with frustrated interactions. We find that the scarred state is stable for perturbations which obey $\mathbb{Z}_2$-symmetry, while it becomes unstable against Heisenberg-type perturbations.\\ \textit{Accepted for publication in Physical Review Research}