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Journal of Physics Conference Series

Numerical Study of the Kitaev-Heisenberg chain as a spin model of the K-intercalated RuCl3

Cliò Efthimia Agrapidis, Jeroen van den Brink, Satoshi Nishimoto

Published Mar 1, 2018• 1 citation
Publisher

Abstract

We study the one-dimensional Kitaev-Heisenberg model as a possible realization of magnetic degrees of freedom of the K-intercalated honeycomb-lattice ruthenium trichloride α -RuCl 3 , denoted as K 0.5 RuCl m . First, we discuss the possible charge ordering patterns in K 0.5 RuCl m , where half of the spins are replaced by nonmagnetic ions in the honeycomb layer. The phase diagram is obtained as a function of nearest-, second-, and third-neighbor repulsions in the atomic limit. We find a strong competition between possible three charge ordering patterns when the repulsions are comparable. We further calculate the spin-spin correlation functions for the 1D Kitaev-Heisenberg model, which is one of the ordering patterns, and consider the effect of the Kitaev term by comparing the results to those for the antiferromagnetic SU(2) Heisenberg chain.

DOI
10.1088/1742-6596/969/1/012112

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We study the one-dimensional Kitaev-Heisenberg model as a possible realization of magnetic degrees of freedom of the K-intercalated honeycomb-lattice ruthenium trichloride α -RuCl 3 , denoted as K 0.5 RuCl m . First, we discuss the possible charge ordering patterns in K 0.5 RuCl m , where half of the spins are replaced by nonmagnetic ions in the honeycomb layer. The phase diagram is obtained as a function of nearest-, second-, and third-neighbor repulsions in the atomic limit. We find a strong competition between possible three charge ordering patterns when the repulsions are comparable. We further calculate the spin-spin correlation functions for the 1D Kitaev-Heisenberg model, which is one of the ordering patterns, and consider the effect of the Kitaev term by comparing the results to those for the antiferromagnetic SU(2) Heisenberg chain.
Honeycomb-lattice magnet α-RuCl 3 is considered to be a potential candidate of realizing Kitaev spin liquid, although this material undergoes a phase transition to the zigzag magnetically ordered state at T N ∼ 7 K. Quite recently, inelastic neutron-scattering experiments using single crystal α-RuCl 3 have unveiled characteristic dynamical properties. We calculate dynamical spin structure factors of three ab-initio models for α-RuCl 3 with an exact numerical diagonalization method. We also calculate temperature dependences of the specific heat by employing thermal pure quantum states. We compare our numerical results with the experiments and discuss characteristics obtained by using three ab-initio models.
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 and φ=3π2 are singular. The ϕ-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.
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.
Recent experiments reveal that the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ is a prime candidate of the Kitaev quantum spin liquid (QSL). However, there is no theoretical model which can properly describe its experimental dynamical response due to the lack of a full understanding of its magnetic interactions. Here, we propose a general scheme to calculate the magnetic interactions in systems (e.g., $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$) with nonnegligible orbital moments by constraining the directions of orbital moments. With this scheme, we put forward a minimal ${J}_{1}\ensuremath{-}{K}_{1}\ensuremath{-}{\mathrm{\ensuremath{\Gamma}}}_{1}\ensuremath{-}{J}_{3}\ensuremath{-}{K}_{3}$ model for $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$ and find that: (I) The third nearest neighbor (NN) antiferromagnetic Heisenberg interaction J3{J}_{3}J3​ stabilizes the zigzag antiferromagnetic order; (II) The NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ plays a pivotal role in determining the preferred direction of magnetic moments and generating the spin wave gap. An exact diagonalization study on this model shows that the Kitaev QSL can be realized by suppressing the NN symmetric off-diagonal exchange ${\mathrm{\ensuremath{\Gamma}}}_{1}$ and the third NN Heisenberg interaction J3{J}_{3}J3​. Thus, we not only propose a powerful general scheme for investigating the intriguing magnetism of Jeff=1/2{J}_{\mathrm{eff}}=1/2Jeff​=1/2 magnets, but also point out future directions for realizing the Kitaev QSL in the honeycomb ruthenium trichloride $\ensuremath{\alpha}\ensuremath{-}\mathrm{RuC}{\mathrm{l}}_{3}$.
Possible effective model with strong Kitaev interaction for α\alphaα-RuCl3{\rm RuCl_3}RuCl3​
We calculate dynamical spin structure factors (DSFs) of three ab-initio models and one ab-initio-guided model for a honeycomb-lattice magnet α\alphaα-RuCl3_33​ with an exact numerical diagonalization method. We also calculate temperature dependences of the heat capacity, the nearest-neighbor (NN) spin-spin correlation function, and the static spin structure factor by employing thermal pure quantum states. From the obtained results for the four effective models, we find that, even when the magnetic order appears at a low temperature, the intensity at the Γ\GammaΓ point in the DSFs grows as the NN spin correlation grows. In addition, we find that the four models fail in explaining the features of both the inelastic-neutron-scattering experiments and the heat capacity measurements. In the four models, when temperature decreases, the heat capacity shows a prominent peak at a high temperature where the nearest-neighbor spin-spin correlation function grows. However, the peak temperatures in the heat capacity are too low in comparison with the observed values in the experiments. Thus, we propose an effective model with the strong ferromagnetic Kitaev coupling. We show that the proposed model quantitatively reproduces the above two experimental features. To discuss the further adequacy of the proposed model, we calculate the field dependence of the polarized terahertz spectra. We find that the proposed model successfully reproduces the experimental results: the spin-gapped excitation surviving up to an onset field where the magnetic order disappears and the almost linear response in the high field region. From the obtained numerical results, we argue that the feature of the low-energy magnetic excitation in α\alphaα-RuCl3_33​ is mainly characterized by the other interactions rather than the most dominant Kitaev interactions such as the off-diagonal interactions on the nearest-neighbor pairs.
We use an exact numerical diagonalization method to calculate the dynamical spin structure factors of three ab initio models and one ab initio guided model for a honeycomb-lattice magnet $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$. We also use thermal pure quantum states to calculate the temperature dependence of the heat capacity, the nearest-neighbor spin-spin correlation function, and the static spin structure factor. From the results obtained from these four effective models, we find that, even when the magnetic order is stabilized at low temperature, the intensity at the $\mathrm{\ensuremath{\Gamma}}$ point in the dynamical spin structure factors increases with increasing nearest-neighbor spin correlation. In addition, we find that the four models fail to explain heat-capacity measurements whereas two of the four models succeed in explaining inelastic-neutron-scattering experiments. In the four models, when temperature decreases, the heat capacity shows a prominent peak at a high temperature where the nearest-neighbor spin-spin correlation function increases. However, the peak temperature in heat capacity is too low in comparison with that observed experimentally. To address these discrepancies, we propose an effective model that includes strong ferromagnetic Kitaev coupling, and we show that this model quantitatively reproduces both inelastic-neutron-scattering experiments and heat-capacity measurements. To further examine the adequacy of the proposed model, we calculate the field dependence of the polarized terahertz spectra, which reproduces the experimental results: the spin-gapped excitation survives up to an onset field where the magnetic order disappears and the response in the high-field region is almost linear. Based on these numerical results, we argue that the low-energy magnetic excitation in $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ is mainly characterized by interactions such as off-diagonal interactions and weak Heisenberg interactions between nearest-neighbor pairs, rather than by the strong Kitaev interactions.