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.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.

arXiv:1202.1610Physical Review BJan 29, 201314 citations

Machine-learned phase diagrams of generalized Kitaev honeycomb magnets

Nihal Rao, Ke Liu, Marc Machaczek +1 more

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--- $J$ , $K$ , 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 ${J}_{3}$ term, whereas the four magnetic orders may compete in particular if ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ is antiferromagnetic.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---$J$, $K$, 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 ${J}_{3}$ term, whereas the four magnetic orders may compete in particular if ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ is antiferromagnetic.

arXiv:2102.01103Physical Review ResearchSep 8, 202118 citations

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.

J

K

, 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

{J}_{3}

term, whereas the four magnetic orders may compete in particular if ${\mathrm{\ensuremath{\Gamma}}}^{\ensuremath{'}}$ is antiferromagnetic.