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arXiv:1202.1610 · Physical Review B

Tuning into the Kitaev spin liquid phase: A spin model on the honeycomb lattice with two types of Heisenberg exchange couplings

Yue Yu, Long Liang, Qian Niu, Shaojing Qin

Published Jan 29, 2013• 14 citations
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Abstract

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 $A$${}_{2}$IrO${}_{3}$ ($A=\text{Li}$, Na). For a monolayer of Li${}_{2}$IrO${}_{3}$, 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.

DOI
10.1103/physrevb.87.041107

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Tuning into the Kitaev spin liquid phase from a new perspective in Li2_22​IrO3_33​
We study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, JJJ and J~\tilde JJ~, where JJJ is for the conventional spin and and J~\tilde JJ~ for rotated spin. When J~=0\tilde J=0J~=0, this is the conventional Heisenberg model. When J=0, the system is either in a stripy antiferromagnetic order(J~0\tilde J 0J~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~J=\tilde JJ=J~). Our model can be applied to layered Mott insulators A2_22​IrO3_33​ (A=Li, Na). For a monolayer of Li2_22​IrO3_33​, 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 study a spin model on honeycomb lattice with two types of Heisenberg exchange couplings, JJJ and J~\tilde JJ~, where JJJ is for the conventional spin and and J~\tilde JJ~ for rotated spin. When J~=0\tilde J=0J~=0, this is the conventional Heisenberg model. When J=0, the system is either in a stripy antiferromagnetic order(J~0\tilde J 0J~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~J=\tilde JJ=J~). Our model can be applied to layered Mott insulators A2_22​IrO3_33​ (A=Li, Na). For a monolayer of Li2_22​IrO3_33​, 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 derive and study a spin one-half Hamiltonian on a honeycomb lattice describing the exchange interactions between Ir4+ ions in a family of layered iridates A2IrO3 (A=Li,Na). Depending on the microscopic parameters, the Hamiltonian interpolates between the Heisenberg and exactly solvable Kitaev models. Exact diagonalization and a complementary spin-wave analysis reveal the presence of an extended spin-liquid phase near the Kitaev limit and a conventional Néel state close to the Heisenberg limit. The two phases are separated by an unusual stripy antiferromagnetic state, which is the exact ground state of the model at the midpoint between two limits.
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We use a recently developed interpretable and unsupervised machine-learning method, the tensorial kernel support vector machine, to investigate the low-temperature classical phase diagram of a generalized Heisenberg-Kitaev-$\mathrm{\ensuremath{\Gamma}}$ ($J\text{\ensuremath{-}}K\text{\ensuremath{-}}\mathrm{\ensuremath{\Gamma}}$) model on a honeycomb lattice. Aside from reproducing phases reported by previous quantum and classical studies, our machine finds a hitherto missed nested zigzag-stripy order and establishes the robustness of a recently identified modulated ${S}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$ phase, which emerges through the competition between the Kitaev and $\mathrm{\ensuremath{\Gamma}}$ spin liquids, against Heisenberg interactions. The results imply that, in the restricted parameter space spanned by the three primary exchange interactions---JJJ, KKK, and $\mathrm{\ensuremath{\Gamma}}$, the representative Kitaev material $\ensuremath{\alpha}\text{\ensuremath{-}}{\mathrm{RuCl}}_{3}$ lies close to the boundaries of several phases, including a simple ferromagnet, the unconventional ${S}_{3}\ifmmode\times\else\texttimes\fi{}{Z}_{3}$, and nested zigzag-stripy magnets. A zigzag order is stabilized by a finite ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ and/or J3{J}_{3}J3​ term, whereas the four magnetic orders may compete in particular if ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ is antiferromagnetic.
We study the ground-state phase diagram of the spin-1/21/21/2 Kitaev-Heisenberg model on the bilayer honeycomb lattice with large-scale tensor network calculations based on the infinite projected entangled pair state technique as well as high-order series expansions. We find that beyond various magnetically ordered phases, including ferromagnetic, zigzag, antiferromagnetic (AFM) and stripy states, two extended quantum spin liquid phases arise in the proximity of the Kitaev limit. While these ordered phases also appear in the monolayer Kitaev-Heisenberg model, our results further show that a valence bond solid state emerges in a relatively narrow range of parameter space between the AFM and stripy phases, which can be adiabatically connected to isolated Heisenberg dimers. Our results highlight the importance of considering interlayer interactions on the emergence of novel quantum phases in the bilayer Kitaev materials.