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arXiv:1903.10026 · Physical Review Letters

One Proximate Kitaev Spin Liquid in the K − J − Γ Model on the Honeycomb Lattice

Jiucai Wang, Bruce Normand, Zheng-Xin Liu

Published Nov 5, 2019• 83 citations
arXivHTMLPDF

Abstract

In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.

DOI
10.1103/physrevlett.123.197201

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In addition to the Kitaev (K) interaction, candidate Kitaev materials also possess Heisenberg (J) and off-diagonal symmetric (Γ) couplings. We investigate the quantum (S=1/2) K-J-Γ model on the honeycomb lattice by a variational Monte Carlo method. In addition to the "generic" Kitaev spin liquid (KSL), we find that there is just one proximate KSL (PKSL) phase, while the rest of the phase diagram contains different magnetically ordered states. The PKSL is a gapless Z_{2} state with 14 Majorana cones, which in contrast to the KSL has a gapless spin response. In a magnetic field applied normal to the honeycomb plane, it realizes two of Kitaev's gapped chiral spin-liquid phases, of which one is non-Abelian with Chern number ν=5 and the other is Abelian with ν=4. These two phases could be distinguished by their thermal Hall conductance.
In this work we study the phase diagram of the Kekul\'e-Kitaev model. The model is defined on a honeycomb lattice with bond-dependent anisotropic exchange interactions, making it exactly solvable in terms of Majorana representation of spins, in close analogy to the Kitaev model. However, the energy spectrum of Majorana fermions has a multiband structure characterized by Chern numbers 0, $\ifmmode\pm\else\textpm\fi{}1$, and $\ifmmode\pm\else\textpm\fi{}2$. We obtained the phase diagram of the model in the plane of exchange couplings and in the presence of a magnetic field and found chiral topological and trivial spin-liquid ground states. In the absence of magnetic field most of the phase diagram is a trivial gapped phase continuously connected to an Abelian phase, while in the presence of the magnetic field a topological phase arises. Furthermore, motivated by recent thermal measurements on the spin-liquid candidate $\ensuremath{\alpha}\ensuremath{-}{\mathrm{RuCl}}_{3}$, we calculated the thermal Hall conductivity at different regimes of parameters and temperatures and found the latter is quantized over a wide range of temperatures.
Variational Monte Carlo (VMC) studies of extended Kitaev honeycomb models reveal a series of multinode quantum spin liquids (QSLs). They have an emergent Z 2 gauge structure, a discrete number of symmetry-protected Majorana cones, and form Abelian or non-Abelian chiral spin liquids in applied magnetic fields. The large number of Z 2 projective symmetry groups for spin–orbit-coupled states on the honeycomb lattice suggests that multinode QSLs can be found experimentally in future work on proximate Kitaev materials.
Recently it was realized that the zigzag magnetic order in Kitaev materials can be stabilized by small negative off-diagonal interactions called the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ terms. To fully understand the effect of the ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ interactions, we investigate the quantum $K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}\text{\ensuremath{-}}{\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ model on the honeycomb lattice using the variational Monte Carlo method. Two multinode Z2{Z}_{2}Z2​ quantum spin liquids (QSLs) are found at ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}>0$, one of which is the previously found proximate Kitaev spin liquid called the PKSL14 state, which shares the same projective symmetry group (PSG) with the Kitaev spin liquid. A remarkable result is that a $\ensuremath{\pi}$-flux state with a distinct PSG appears at larger ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$. The $\ensuremath{\pi}$-flux state is characterized by an enhanced periodic structure in the spinon dispersion in the original Brillouin zone (BZ), which is experimentally observable. Interestingly, two PKSL8 states are competing with the $\ensuremath{\pi}$-flux state and one of them can be stabilized by six-spin ring-exchange interactions. The physical properties of these nodal QSLs are studied by applying magnetic fields and the results depend on the number of cones. Our study infers that there exist a family of zero-flux QSLs that contain $6n+2,n\ensuremath{\in}\mathbb{Z}$ Majorana cones and a family of $\ensuremath{\pi}$-flux QSLs containing 4(6n+2)4(6n+2)4(6n+2) cones in the original BZ. It provides guidelines for experimental realization of non-Kitaev QSLs in relevant materials.
Motivated by recent experimental observations in α-RuCl_{3}, we study the K-Γ model on the honeycomb lattice in an external magnetic field. By a slave-particle representation and variational Monte Carlo calculations, we reproduce the phase transition from zigzag magnetic order to a field-induced disordered phase. The nature of this state depends crucially on the field orientation. For particular field directions in the honeycomb plane, we find a gapless Dirac spin liquid, in agreement with recent experiments on α-RuCl_{3}. For a range of out-of-plane fields, we predict the existence of a Kalmeyer-Laughlin-type chiral spin liquid, which would show an integer-quantized thermal Hall effect.
By including a material-relevant off-diagonal interaction called the $\mathrm{\ensuremath{\Gamma}}$ term into the Kitaev model and introducing spatial anisotropy in the interaction strength on the honeycomb lattice, we obtain a series of nodal Z2{Z}_{2}Z2​ quantum spin liquids (QSLs) from the parton approach. These QSLs share the same projective symmetry group and are characterized by certain numbers of symmetry-protected Majorana cones in their low-energy excitation spectrum. We illustrate that the physical properties of the QSLs are dependent on the information of the cones. Using the $\mathbit{k}\ifmmode\cdot\else\textperiodcentered\fi{}\mathbit{p}$ method, we analyze the chirality of every cone with respect to mass generating perturbations. Especially, for an applied external magnetic field, we provide the maximum-mass field orientation for every cone. Thus, for arbitrarily oriented weak magnetic fields, we can immediately read out the Chern number of the system and the properties of the resultant chiral spin liquids. The new gapless QSLs predicted in our phase diagrams are promising to be realized experimentally by exerting uniaxial pressure to tune the anisotropy of the interaction strength. We further show that all these QSLs can be distinguished by measurable quantities. Based on the study of these QSL phases, we conclude that a complete classification of nodal QSLs with certain symmetry should include not only the projective symmetry groups, but also the information of the cones, i.e., their total number, their chiralities, and the way in which they are symmetry related.
Temperature evolution of the spin correlation and excitation spectrum is investigated for the Kitaev model defined on a decorated honeycomb lattice by using the quantum Monte Carlo simulation in the Majorana fermion representation. The ground state of this quantum spin model is given by two kinds of chiral spin liquids: one is topologically trivial with Abelian anyon excitations, and the other is topologically nontrivial with non-Abelian anyon excitations. While lowering temperature, the model exhibits several crossovers in the paramagnetic state, which originate from the fractionalization of quantum spins into Majorana fermions, in addition to a phase transition associated with time reversal symmetry breaking. We show that the spin correlation develops around the crossover temperatures, whereas it shows a slight change at the critical temperature, as in other Kitaev-type models. We also calculate the excitation spectrum in terms of Majorana fermions, and find that the excitation gap in the non-Abelian phase is fragile against thermal fluctuations of the Z 2 fluxes, while that in the Abelian phase remains open.