Abstract Aligning with the ongoing search for quantum spin liquids (QSLs), identifying QSLs in Kitaev magnets has attracted significant research interest during the past decade. Nevertheless, it remains a significant challenge. One of the major difficulties is that Kitaev QSL is typically fragile to competing interactions such as off-diagonal exchanges, which are ubiquitous in real materials owing to spin-orbit coupling and crystal-field effects. This, in turn, generates many intriguing field-induced novel phases and the thermal Hall effect. In this review, we focus on the interplay between the Kitaev interaction and off-diagonal Γ and Γ′ exchange from a numerical perspective. The review discusses certain representative exotic phases such as Γ spin liquid, nematic ferromagnet, spin-flop phase, and distinct chiral spin states with spontaneous time-reversal symmetry breaking. It also presents quantum phase diagrams of anisotropic Kitaev-Γ chains that exhibit kaleidoscopy for both ordered and disordered phases.
Abstract Aligning with the ongoing search for quantum spin liquids (QSLs), identifying QSLs in Kitaev magnets has attracted significant research interest during the past decade. Nevertheless, it remains a significant challenge. One of the major difficulties is that Kitaev QSL is typically fragile to competing interactions such as off-diagonal exchanges, which are ubiquitous in real materials owing to spin-orbit coupling and crystal-field effects. This, in turn, generates many intriguing field-induced novel phases and the thermal Hall effect. In this review, we focus on the interplay between the Kitaev interaction and off-diagonal Γ and Γ′ exchange from a numerical perspective. The review discusses certain representative exotic phases such as Γ spin liquid, nematic ferromagnet, spin-flop phase, and distinct chiral spin states with spontaneous time-reversal symmetry breaking. It also presents quantum phase diagrams of anisotropic Kitaev-Γ chains that exhibit kaleidoscopy for both ordered and disordered phases.
The higher-spin Kitaev magnets, in which the Kitaev interaction and off-diagonal exchange couplings are overwhelmingly large, have emerged as a fertile avenue to explore exotic phases and unusual excitations. In this paper, we study the quantum phase diagram of the spin-1 Kitaev-$\mathrm{\ensuremath{\Gamma}}$ model on the honeycomb lattice using density-matrix renormalization group. It harbors six distinct phases and the intriguing findings are three magnetically ordered phases in which both time-reversal symmetry and lattice symmetry, albeit of different sorts, are broken spontaneously. The chiral spin state originates from the order-by-disorder effect and exhibits an almost saturated scalar spin chirality at the quantum level. Depending on the relative strength of the two interactions, it also features a columnarlike or plaquettelike dimer pattern as a consequence of the translational symmetry breaking. In parallel, the nematic ferromagnets are situated at the ferromagnetic Kitaev side and possess small but finite ferromagnetic ordering. The lattice-rotational symmetry breaking enforces nonequivalent bond energy along one of the three bonds. Although the intrinsic difference between the two nematic ferromagnets remains elusive, the discontinuities in the von Neumann entropy, hexagonal plaquette operator, and Wilson loop operator convincingly suggest that they are separated via a first-order phase transition.
The possible existence of a quantum spin liquid (QSL) phase, an exotic state of matter with long-range quantum entanglement and fractionalized excitations, in α-RuCl3 has sparked widespread interests in exploring QSLs in various Kitaev models under magnetic fields. Recently, a K-J-Γ-Γ′ model has been proposed to accurately describe the compound by fitting the measured thermodynamic data, where J is Heisenberg interaction, and Γ, Γ′ are off-diagonal exchanges on top of the dominant Kitaev coupling K. Based on this effective model, an intermediate QSL phase in presence of an out-of-plane fields along the [111] direction, between the low-field zigzag order and high-field polarized phase, has been predicted. By combining density matrix renormalization group (DMRG), exponential tensor renormalization group (XTRG), and variational Monte Carlo (VMC) calculations, we address the nature of this QSL phase in the honeycomb K-J-Γ-Γ′ model under the [111]-direction field. Our DMRG calculations find the algebraic-like decay of spin correlation function, the finite spin scalar chiral order, and lattice nematic order. Together with the XTRG results of power-law specific heat at low temperature, our findings naturally suggest a gapless nematic chiral spin liquid. On the other hand, our VMC study finds a gapped nematic chiral spin liquid with the variational energy very close to that obtained in DMRG. As VMC finds a very small gap that is beyond the current resolution of both ground-state DMRG and finite-temperature XTRG calculations on finite-width cylinders, we resort the full clarification for the nature of the intermediate QSL to future studies. Lastly, we discuss the implications of our results to the recent experiment on α-RuCl3 and the QSL-like phase in a similar K-Γ-Γ′ model.
We explore the phase diagram of the Kitaev-Heisenberg model with nearest neighbor interactions on the honeycomb lattice using the exact diagonalization of finite systems combined with the cluster mean field approximation, and supplemented by the insights from analytic approaches: the linear spin-wave and second-order perturbation theories. This study confirms that by varying the balance between the Heisenberg and Kitaev term, frustrated exchange interactions stabilize in this model either one of four phases with magnetic long range order: N\'eel phase, ferromagnetic phase, and two other phases with coexisting antiferromagnetic and ferromagnetic bonds, zigzag and stripy phase, or one of two distinct spin-liquid phases. Out of these latter disordered phases, the one with ferromagnetic Kitaev interactions has a substantially broader range of stability as the neighboring competing ordered phases, ferromagnetic and stripy, have very weak quantum fluctuations. Focusing on the quantum spin-liquid phases, we study spatial spin correlations and dynamic spin structure factor of the model by the exact diagonalization technique, and discuss the evolution of gapped low-energy spin response across the quantum phase transitions between the disordered spin liquid and phases with long range magnetic order.
Kitaev magnets have emerged as pivotal systems for investigating frustrated magnetism, providing a unique platform to explore quantum phases governed by the interplay between bond-dependent anisotropy and external magnetic fields. However, the quantum phase diagrams, particularly near the dominant antiferromagnetic Kitaev regime, remain puzzling despite extensive studies. In this work, we perform unbiased exact diagonalization calculations of the Kitaev-Γ model in a [111] magnetic field on a C6-symmetric 24-site cluster. By calculating the Z2 flux density and the topological entanglement entropy, we reveal multiple phase transitions and identify signatures of both scalar and vector chiral orders in the intermediate-field regime between the Kitaev spin liquid and the polarized phase. As the negative Γ interaction increases, we discover a proximate quantum spin liquid featured by a three-peak specific heat and a spin-flop phase at a moderate magnetic field. Our findings provide insight into the field-induced intermediate phases in the antiferromagnetic Kitaev model and pave the way for the hunt for emergent phases in real materials.
Abstract 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.
Abstract Finding the Kitaev spin liquid in candidate materials involves understanding the entire phase diagram, including other allowed interactions. One of these interactions, called the Gamma (Γ) interaction, causes magnetic frustration and its interplay with the Kitaev (K) interaction is crucial to comprehend Kitaev materials. Due to the complexity of the combined KΓ model, quasi-one-dimensional models have been investigated. While several disordered phases are found in the 2-leg ladder, the nature of the phases are yet to be determined. Here we focus on the disordered phase near the antiferromagnetic Γ limit (denoted by AΓ phase) next to the ferromagnetic Kitaev phase. We report a distinct non-local string order parameter characterizing the AΓ phase, different from the string order parameter in the Kitaev phase. This string order parameter becomes evident only after two unitary transformation, referred to as a twice hidden string order parameter. The related entanglement spectrum, edge states, magnetic field responses, and the symmetry protecting the phase are presented, and its relevance to the two-dimensional Kitaev materials is discussed. Two newly identified disordered phases in the phase diagram of KΓ ladder is also reported.