The microscopic origin of the topological magnon band gap in ${\mathrm{CrI}}_3$ ferromagnets has been a subject of controversy for years since two main models with distinct characteristics, i.e., Dzyaloshinskii-Moriya (DM) and Kitaev, provided possible explanations with different outcome implications. Here, we investigate the angular magnetic field dependence of the magnon gap of ${\mathrm{CrI}}_3$ by elucidating what main contributions play a major role in its generation. We implement stochastic atomistic spin-dynamics simulations to compare the impact of these two spin interactions on the magnon spectra. We observe three distinct magnetic field dependencies between these two gap opening mechanisms. First, we demonstrate that the Kitaev-induced magnon gap is influenced by both the direction and amplitude of the applied magnetic field, while the DM-induced gap is solely affected by the magnetic field direction. Second, the position of the Dirac cones within the Kitaev-induced magnon gap shifts in response to changes in the magnetic field direction, whereas they remain unaffected by the magnetic field direction in the DM-induced gap scenario. Third, we find a direct-indirect magnon band gap transition in the Kitaev model by varying the applied magnetic field direction. These differences may distinguish the origin of topological magnon gaps in ${\mathrm{CrI}}_3$ and other van der Waals magnetic layers. Our findings pave the way for exploration and engineering topological gaps in van der Waals materials.

中文翻译：

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范德华 CrI3 铁磁体中的拓扑磁振子能隙工程

拓扑磁振子带隙的微观起源${\mathrm{肌酐}}_3$铁磁体多年来一直是争议的话题，因为两个具有不同特征的主要模型，即 Dzyaloshinskii-Moriya (DM) 和 Kitaev，提供了具有不同结果含义的可能解释。在这里，我们研究了磁振子间隙的角磁场依赖性${\mathrm{肌酐}}_3$通过阐明哪些主要贡献在其产生过程中发挥了重要作用。我们实施随机原子自旋动力学模拟来比较这两种自旋相互作用对磁振子谱的影响。我们观察到这两种间隙打开机制之间的三种不同的磁场依赖性。首先，我们证明 Kitaev 感应磁振子间隙受到施加磁场的方向和幅度的影响，而 DM 感应间隙仅受磁场方向影响。其次，基塔耶夫引起的磁振子间隙内狄拉克锥的位置会随着磁场方向的变化而发生变化，而在 DM 引起的间隙情况下，它们不受磁场方向的影响。第三，我们通过改变施加的磁场方向，在 Kitaev 模型中发现了直接-间接磁振子带隙转变。这些差异可以区分拓扑磁振子间隙的起源${\mathrm{肌酐}}_3$和其他范德华磁性层。我们的发现为探索和工程范德华材料的拓扑间隙铺平了道路。