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arXiv:2403.14754 · Physical Review Letters

Transforming from Kitaev to Disguised Ising Chain: Application to CoNb 2 O 6

Derek Churchill, Hae‐Young Kee

Published Jul 30, 2024• 12 citations
arXivHTMLPDF

Abstract

For many years, CoNb_{2}O_{6} has served as an exemplar of the one-dimensional Ising model. However, recent experimental and theoretical analyses challenge its applicability to this material. Prior to that, a tailored spin model for 3d^{7} systems such as Co^{2+}, known as the JKΓ model, has emerged, featuring Heisenberg (J), Kitaev (K), and Gamma (Γ) interactions. While these interactions are permitted by the symmetry of the system, their role in CoNb_{2}O_{6} remains enigmatic. We present a microscopic theory based on spin-orbit entangled J_{eff}=1/2 states, aimed at elucidating the roles of Kitaev and Gamma interactions in shaping Ising anisotropy. Leveraging strong coupling theory, we identify a dominant ferromagnetic Kitaev interaction. Furthermore, by comparing dynamical structure factors obtained via exact diagonalization with those from inelastic neutron scattering experiments, we find an antiferromagnetic Γ interaction, which dictates the Ising axis and explains the mechanism behind moment pinning. Our theory provides a microscopic origin for Ising behavior in spin-orbit coupled one-dimensional chains and posits CoNb_{2}O_{6} as a rare Kitaev chain.

DOI
10.1103/physrevlett.133.056703

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For many years, CoNb_{2}O_{6} has served as an exemplar of the one-dimensional Ising model. However, recent experimental and theoretical analyses challenge its applicability to this material. Prior to that, a tailored spin model for 3d^{7} systems such as Co^{2+}, known as the JKΓ model, has emerged, featuring Heisenberg (J), Kitaev (K), and Gamma (Γ) interactions. While these interactions are permitted by the symmetry of the system, their role in CoNb_{2}O_{6} remains enigmatic. We present a microscopic theory based on spin-orbit entangled J_{eff}=1/2 states, aimed at elucidating the roles of Kitaev and Gamma interactions in shaping Ising anisotropy. Leveraging strong coupling theory, we identify a dominant ferromagnetic Kitaev interaction. Furthermore, by comparing dynamical structure factors obtained via exact diagonalization with those from inelastic neutron scattering experiments, we find an antiferromagnetic Γ interaction, which dictates the Ising axis and explains the mechanism behind moment pinning. Our theory provides a microscopic origin for Ising behavior in spin-orbit coupled one-dimensional chains and posits CoNb_{2}O_{6} as a rare Kitaev chain.
Abstract Motivated by the on-going discussion on the nature of magnetism in the quantum Ising chain CoNb 2 O 6 , we present a first-principles-based analysis of its exchange interactions with additional modeling, addressing drawbacks of a purely density functional theory ansatz. This method allows us to extract and understand the origin of the magnetic couplings—including all symmetry-allowed terms - and resolve conflicting model descriptions in CoNb 2 O 6 . We find that the twisted Kitaev chain and transverse-field ferromagnetic Ising chain views are mutually compatible, although additional off-diagonal exchanges are required for a complete picture. We show that the dominant exchange interaction is a ligand-centered process—involving e g electrons -, rendered anisotropic by low-symmetry crystal fields in CoNb 2 O 6 , resulting in dominant Ising exchange. Smaller bond-dependent anisotropies are found to originate from d − d kinetic exchange processes involving t 2 g electrons. We demonstrate the validity of our low-energy model by comparing its predictions to measured THz and INS spectra.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interaction emerges from the interplay between the Kitaev ( K ) and Heisenberg ( J ) exchanges, and is accompanied by an additional bond-dependent interaction (Γ), leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J - K -Γ model with four-fold screw symmetry in a spin-orbit-entangled chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic ordered structures, the transition remains in the Ising universality class with central charge c = 1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to Ising materials such as chains of Co 2+ ions is also discussed.
We calculate the dynamical spin structure factor of the generalized spin-1/21/21/2 compass spin chain using the density matrix renormalization group. The model, also known as the twisted Kitaev spin chain, was recently proposed to be relevant for the description of the spin chain compound CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​. It features bond-dependent interactions and interpolates between an Ising chain and a one-dimensional variant of Kitaev's honeycomb spin model. The structure factor, in turn, is found to interpolate from gapped and nondispersive in the Ising limit to gapless with nontrivial continua in the Kitaev limit. In particular, the component of the structure factor perpendicular to the Ising directions changes abruptly at the Kitaev point into a dispersionless continuum due to the emergence of an extensive ground-state degeneracy. We show this continuum is consistent with analytical Jordan-Wigner results. We also discuss implications for future inelastic scattering experiments and applications to materials, particularly CoNb2O6{\mathrm{CoNb}}_{2}{\mathrm{O}}_{6}CoNb2​O6​.
Evolution from Kitaev to XXZ spin chains via distortion: Application to BaCo2_22​V2_22​O8_88​
BaCo2_22​V2_22​O8_88​ is a prototypical spin-orbit-coupled Ising-chain antiferromagnet that provides a unique platform for studying field-induced quantum magnetism. Under a transverse magnetic field, it exhibits unusual magnetic properties, including an anomalous staggered magnetization and a pronounced anisotropy of the critical field with respect to the in-plane field direction. While these phenomena have been attributed phenomenologically to a site-dependent anisotropic ggg-tensor, a recent microscopic theory has shown that spin-orbit coupling naturally generates bond-dependent Heisenberg, Kitaev, and ΓΓΓ exchange interactions. Here, we unify these two pictures by extending the microscopic theory to incorporate distortions of the CoO6_66​ octahedra. We show that the distortions not only generate the site-dependent anisotropic ggg-tensor but also renormalize the staggered exchange interactions through a distortion-induced contribution that partially compensates the Kitaev-derived staggered term. Using point-charge calculations to estimate the ggg-tensor of BaCo2_22​V2_22​O8_88​, we demonstrate that the strong anisotropy of the critical field originates from the combined effects of the modified exchange interactions and the anisotropic ggg-tensor. Our work provides a unified microscopic framework for understanding the magnetic anisotropy of spin-orbit-coupled Ising-chain materials.
Abstract The Kitaev spin liquid, a ground state of the bond-dependent Kitaev model in a honeycomb lattice has been a center of attraction, since a microscopic theory to realize such an interaction in solid-state materials was discovered. A challenge in real materials though is the presence of the Heisenberg and another bond-dependent Gamma interactions detrimental to the Kitaev spin liquid, and there have been many debates on their relative strengths. Here we offer a strategy to extract the Kitaev interaction out of a full microscopic model by utilizing the symmetries of the Hamiltonian. Two tilted magnetic field directions related by a two-fold rotational symmetry generate distinct spin excitations originated from a specific combination of the Kitaev and Gamma interactions. Together with the in- and out-of-plane magnetic anisotropy, one can determine the Kitaev and Gamma interactions separately. Dynamic spin structure factors are presented to motivate future experiments. The proposed setups will advance the search for Kitaev materials.
The one-dimensional transverse Ising model is a paradigmatic example of quantum criticality. In spin-orbit coupled systems, however, effective Ising interactions arise alongside bond-dependent couplings such as Kitaev (KKK) and ΓΓΓ terms in addition to the Heisenberg (JJJ) interaction, leading to complex magnetic orders beyond the pure Ising limit. We first explore how the generic J−K−ΓJ-K-ΓJ−K−Γ model with four-fold screw symmetry in a spin-orbit-entangled pseudospin-1/2 chain manifests via sublattice order, and then test whether field-driven transitions retain Ising universality. We find sublattice-dependent magnetic order below the critical field and a distinct sublattice pattern persisting above it. Despite these complex magnetic order structures, the transition remains in the Ising universality class with central charge c=1/2c=1/2c=1/2. Our work provides a route to the microscopic Hamiltonian and emergent Ising criticality while allowing microscopic physics of the sublattice orders to manifest at low and high fields. Application to antiferromagnetic Ising materials such as BaCo2_22​V2_22​O8_88​ is also discussed.