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arXiv:1909.10170 · International Conference on Strongly Correlated Electron Systems (SCES)

Magnetic Properties of the S = 1 Kitaev Model with Anisotropic Interactions

Tetsuya Minakawa, Joji Nasu, Akihisa Koga

Published Mar 18, 2020• 0 citations
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Abstract

We investigate magnetic properties in the $S=1$ Kitaev model in the anisotropic limit. Performing the fourth-order perturbation expansion with respect to the $x$-bonds, $y$-bonds, and magnetic field, we derive the effective Hamiltonian, where the low-energy physics should be described by the free spins with an effective magnetic field. Making use of the exact diagonalization method for small clusters, we discuss ground-state properties in the system complementary.

DOI
10.7566/jpscp.30.011086

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We investigate magnetic properties in the S=1S=1S=1 Kitaev model in the anisotropic limit. Performing the fourth-order perturbation expansion with respect to the xxx-bonds, yyy-bonds, and magnetic field, we derive the effective Hamiltonian, where the low-energy physics should be described by the free spins with an effective magnetic field. Making use of the exact diagonalization method for small clusters, we discuss ground-state properties in the system complementary.
We study low-energy properties of spin-SSS Kitaev models in an anisotropic limit. The effective form of a local conserved quantity is derived in the low-energy subspace. We find this is the same as that of S=1/2S=1/2S=1/2 case for the half-integer spins but shows a different form for the integer spins. Applying the perturbation theory to the anisotropic Kitaev model, we obtain the effective Hamiltonian. In the integer spin case, the effective model is equivalent to a free spin model under a uniform magnetic field, where quantum fluctuations are quenched. On the other hand, in the half-integer case, the system is described by the toric code Hamiltonian, where quantum fluctuations play a crucial role in the ground state. The boundary effect in the anisotropic Kitaev system is also discussed.
Recently, there has been a renewed interest in properties of the higher-spin Kitaev models, especially their low-dimensional analogs with additional interactions. These quasi-one-dimensional systems exhibit rich phase diagrams with symmetry-protected topological phases, Luttinger liquids, hidden order, and higher-rank magnetism. However, the nature of the pure spin-SSS Kitaev chains is not yet fully understood. Earlier works found a unique ground state with short-ranged correlations for S=1S=1S=1 and an intriguing double-peak structure in the heat capacity associated with an entropy plateau. To understand the low-energy excitations and thermodynamics for general SSS, we study the anisotropic spin-SSS Kitaev chain. Starting from the dimerized limit, we derive an effective low-energy Hamiltonian at finite anisotropy. For half-integer spins we find a trivial effective model, reflecting a nonlocal symmetry protecting the degeneracy, while for integer SSS we find interactions among the flux degrees of freedom that select a unique ground state. The effective model for integer spins is used to predict the low-energy excitations and thermodynamics, and we make a comparison with the semiclassical limit through linear spin wave theory. Finally, we speculate on the nature of the isotropic limit.
Motivated by intense research on two-dimensional spin-12\frac{1}{2}21​ Kitaev materials, Kitaev spin chains and ladders, though geometrically limited, have been studied for their numerical simplicity and insights into extended Kitaev models. The phase diagrams under the magnetic field were also explored for these quasi-one-dimensional models. For an isotropic Kitaev chain, it was found that a magnetic field polarizes the ground state except along the symmetric field angle, where the chain is found to remain gapless up to a critical field strength where it enters an intriguing soliton phase before reaching the polarized state at higher field strengths. Here we study an anisotropic Kitaev chain under a magnetic field using the density matrix renormalization group technique, where the ground state has a macroscopic degeneracy with a finite gap in the absence of the magnetic field. When the field is mainly aligned parallel to the strong bond, four-site and large unit-cell ordered phases arise. In a certain angle of the field, another ordered phase characterized by a uniform chirality with six-site periodicity emerges. We employ a perturbation theory to understand such field-induced ordered phases. The effective model uncovers the presence of transverse Ising and Dzyaloshinskii-Moriya interactions between unit cells, as well as further-neighbor Ising interaction induced by the magnetic field, which collectively explain the mechanisms behind these ordered states. Open questions and challenges are also discussed.
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Magnetic phases with quantum entanglement are often expressed in terms of parton wavefunctions. Relatively few examples are known where wavefunctions can be directly written down in the spin basis. In this article, we consider the spin-SSS Kitaev model in one dimension. For S=1/2S=1/2S=1/2, its eigenstates can be written using a Jordan-Wigner fermionic representation. Here, we present ground state wavefunctions for any SSS directly in the spin basis. The states we propose are valence bond arrangements, with bonds having singlet or triplet character for S=1/2S=1/2S=1/2. For S>1/2S>1/2S>1/2, we use bond-states that serve as analogues of singlets and triplets. We establish the validity of our wavefunctions using a perturbative approach starting from an anisotropic limit, with key features surviving to all orders in perturbation theory. For half-integer SSS and periodic boundaries, we have exponential ground state degeneracy. The ground states are subject to a nonlocal constraint. They have `triplets' superposed on a background of singlets, but with the total number of triplets constrained to be even. For integer SSS, a unique ground state emerges, composed purely of `triplets'. Our spin-basis wavefunctions, while not exact, capture the dominant weight of the ground state(s). We obtain good agreement against exact diagonalization wavefunctions and Jordan-Wigner spectra.
The exact solution of Kitaev's spin-1/21/21/2 honeycomb spin-liquid model has sparked an intense search for Mott insulators hosting bond-dependent Kitaev interactions, of which Na2IrO3{\mathrm{Na}}_{2}{\mathrm{IrO}}_{3}Na2​IrO3​ and $\ensuremath{\alpha}\text{\ensuremath{-}}\mathrm{Ru}{\mathrm{Cl}}_{3}$ are prime examples. Subsequently, it has been proposed that also spin-1 and spin-3/23/23/2 analogs of Kitaev interactions may occur in materials with strong spin-orbit coupling. As a minimal model to describe these Kitaev materials, we study the Heisenberg-Kitaev Hamiltonian in a consistent 1/S1/S1/S expansion, with SSS being the spin size. We present a comprehensive study of this model in the presence of an external magnetic field applied along two different directions, [001] and [111], for which an intricate classical phase diagram has been reported. In both settings, we employ spin-wave theory in a number of ordered phases to compute phase boundaries at the next-to-leading order in 1/S1/S1/S and show that quantum corrections substantially modify the classical phase diagram. More broadly, our work presents a consistent route to investigate the leading quantum corrections in spin models that break spin-rotational symmetry.