Motivated by systems that can be seen as composed of two frustratedsublattices combined into a less frustrated total lattice, we study thedouble-exchange model with nearest-neighbor (NN) and next--nearest-neighbor(NNN) couplings on the honeycomb lattice. When adding NN hopping and itsresulting double exchange to the antiferromagnetic (AFM) Heisenberg coupling,the resulting phase diagram is quite different from that of purelyHeisenberg-like magnetic models and strongly depends on electron filling. Forhalf filling, patterns of AFM dimers dominate, where the effective electronicbands remain graphene-like with Dirac cones in all phases, from the FM to the$120^\circ$ limit. When the density of states at the Fermi level is sizable, wefind non-coplanar incommensurate states as well as a small-vortex phase.Finally, a non-coplanar commensurate pattern realizes a Chern insulator atquarter filling. In the case of both NN and NNN hopping, the noncoplanar spinpattern inducing Chern insulators in triangular lattices is found to be quitestable under coupling into a honeycomb system. The resulting total phases aretopologically nontrivial and either a Chern insulator with $C=2$ or a magnetictopological crystalline insulator protected by a combination ormirror-reflection and time-reversal symmetries arise.
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机译：受可被视为由两个沮丧的子晶格组合成较少沮丧的总晶格组成的系统的激励，我们研究了蜂窝晶格上具有最近邻居（NN）和次邻邻居（NNN）耦合的双交换模型。当在反铁磁（AFM）Heisenberg耦合中添加NN跳变并导致其双重交换时，所产生的相图与纯Heisenberg类磁模型的相图完全不同，并且在很大程度上取决于电子填充。对于半填充，AFM二聚体的模式占主导地位，在有效相带中，从FM到$ 120 ^ \ circ的所有相位，狄拉克锥都与石墨烯类似。当费米能级的状态密度足够大时，我们确定非共面不相称状态以及小涡旋相。最后，非共面相称模式实现了Chern绝缘子的四角填充。在NN和NNN跳变的情况下，发现在耦合到蜂窝系统中时，三角晶格中非共面自旋图案诱导的Chern绝缘子非常稳定。所得的总相在拓扑上是无关紧要的，并且会出现由$ C = 2 $构成的Chern绝缘子或由镜面反射和时间反转对称性组合保护的磁拓扑晶体绝缘子。
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