The exactly solvable Kitaev honeycomb lattice model is realized as the low-energy effect Hamiltonian of a spin-1/2 model with spin rotation and time-reversal symmetry. The mapping to low-energy effective Hamiltonian is exact without truncation errors in traditional perturbation series expansions. This model consists of a honeycomb lattice of clusters of four spin-1/2 moments and contains short-range interactions up to six-spin (or eight-spin) terms. The spin in the Kitaev model is represented not as these spin-1/2 moments but as pseudospin of the two-dimensional spin-singlet sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin correlations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this model. This exact construction is quite general and can be used to make other interesting spin-1/2 models from spin-rotation invariant Hamiltonians. We discuss two possible routes to generate the high-order spin interactions from more natural couplings, which involves perturbative expansions thus breaks the exact mapping, although in a controlled manner.
The exactly solvable Kitaev honeycomb lattice model is realized as the low-energy effect Hamiltonian of a spin-1/2 model with spin rotation and time-reversal symmetry. The mapping to low-energy effective Hamiltonian is exact without truncation errors in traditional perturbation series expansions. This model consists of a honeycomb lattice of clusters of four spin-1/2 moments and contains short-range interactions up to six-spin (or eight-spin) terms. The spin in the Kitaev model is represented not as these spin-1/2 moments but as pseudospin of the two-dimensional spin-singlet sector of the four antiferromagnetically coupled spin-1/2 moments within each cluster. Spin correlations in the Kitaev model are mapped to dimer correlations or spin-chirality correlations in this model. This exact construction is quite general and can be used to make other interesting spin-1/2 models from spin-rotation invariant Hamiltonians. We discuss two possible routes to generate the high-order spin interactions from more natural couplings, which involves perturbative expansions thus breaks the exact mapping, although in a controlled manner.
We analyze the gapped phase of the Kitaev honeycomb model perturbatively in the isolated-dimer limit. Our analysis is based on the continuous unitary transformations method, which allows one to compute the spectrum as well as matrix elements of operators between eigenstates at high order. The starting point of our study consists of an exact mapping of the original honeycomb spin system onto a square-lattice model involving an effective spin and a hard-core boson. We then derive the low-energy effective Hamiltonian up to order 10 which is found to describe an interacting-anyon system, contrary to the order 4 result which predicts a free theory. These results give the ground-state energy in any vortex sector and thus also the vortex gap, which is relevant for experiments. Furthermore, we show that the elementary excitations are emerging free fermions composed of a hard-core boson with an attached spin- and phase-operator string. We also focus on observables and compute, in particular, the spin-spin correlation functions. We show that they admit a multiplaquette expansion that we derive up to order 6. Finally, we study the creation and manipulation of anyons with local operators, show that they also create fermions, and discuss the relevance of our findings for experiments in optical lattices.
The exactly solvable spin-1/2 Kitaev model on a honeycomb lattice has drawn significant interest, as it offers a pathway to realizing the long-sought-after quantum spin liquid. Building upon the Kitaev model, Yao and Lee introduced another exactly solvable model on an unusual star lattice featuring non-abelian spinons. The additional pseudospin degrees of freedom in this model could provide greater stability against perturbations, making this model appealing. However, a mechanism to realize such an interaction in a standard honeycomb lattice remains unknown. Here, we provide a microscopic theory to obtain the Yao-Lee model, on a honeycomb lattice by utilizing strong spin-orbit coupling of anions edge-shared between two e g ions in the exchange processes. This mechanism leads to the desired bond-dependent interaction among spins rather than orbitals, unique to our model, implying that the orbitals fractionalize into gapless Majorana fermions and fermionic octupolar excitations emerge. Since the conventional Kugel-Khomskii interaction also appears, we examine the phase diagram, including these interactions, using classical Monte Carlo simulations and exact diagonalization techniques. Our findings reveal a broad region of disordered states that break rotational symmetry in the bond energy, suggesting intriguing behavior reminiscent of a spin-orbital liquid.
We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, J and J~, where J is for the conventional spin and and J~ for rotated spin. When J~=0, this is the conventional Heisenberg model. When J=0, the system is either in a stripy antiferromagnetic order(J~0, antiferromagnetic for rotated spin). The competition between two ferromagnetic orders or two antiferromagnetic orders induces Kitaev's spin liquid phase characterized by the exactly solvable Kitaev model (J=J~). Our model can be applied to layered Mott insulators A2IrO3 (A=Li, Na). For a monolayer of Li2IrO3, we show that it is possible to tune the controlling parameter into the Kitaev spin liquid regime by a link-dependent Rashba spin-orbital coupling.
We introduce a spin-21 model in three dimensions which is a generalization of the well-known Kitaev model on a honeycomb lattice. Following Kitaev, we solve the model exactly by mapping it to a theory of noninteracting fermions in the background of a static Z2 gauge field. The phase diagram consists of a gapped phase and a gapless one, similar to the two-dimensional case. Interestingly, unlike in the two-dimensional model, in the gapless phase the gap vanishes on a contour in the k space. Furthermore, we show that the flux excitations of the gauge field, due to some local constraints, form looplike structures; such loops exist on a lattice formed by the plaquettes in the original lattice and is topologically equivalent to the pyrochlore lattice. Finally, we derive a low-energy effective Hamiltonian that can be used to study the properties of the excitations in the gapped phase.
We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, J and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$, where J is for the conventional spin and $\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$ is for rotated spin. When J=0, the system is either in a stripy antiferromagnetic order ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}<0$, ferromagnetic for rotated spin) or a zigzag antiferromagnetic order ($\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}>0$, antiferromagnetic for rotated spin). The competition between two ferromagnetic orders or between two antiferromagnetic orders induces Kitaev's spin liquid phase, characterized by the exactly solvable Kitaev model ($J=\stackrel{\ifmmode \tilde{}\else \~{}\fi{}}{J}$). Our model can be applied to layered Mott insulators A2IrO3 (A=Li, Na). For a monolayer of Li2IrO3, we show that it is possible to tune the controlling parameter into the Kitaev spin liquid regime by a link-dependent Rashba spin-orbital coupling.
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.