The topological properties of non-Hermitian Hamiltonian is a hot topic, and the theoretical studies along this research line are usually based on the two-level non-Hermitian Hamiltonian (or, equivalently, a spin-$1/2$ non-Hermitian Hamiltonian). We are motivated to study the geometrical phases of a three-level Lieb lattice model (or, equivalently, a spin-$1$ non-Hermitian Hamiltonian) with the flat band in the context of a polariton condensate. The topological invariants are calculated by both winding numbers in the Brillouin zone and the geometrical phase of Majorana stars in the Bloch sphere. Besides, we provide an intuitive way to study the topological phase transformation with the higher spin, and the flat band offers a platform to define the topological phase transition on the Bloch sphere. According to the trajectories of the Majorana stars, we calculate the geometrical phases of the Majorana stars. We study the Lieb lattice with a complex hopping and find their phases have a jump when the parameters change from the trivial phase to the topological phase. The correlation phase of Majorana stars will rise along with the increase of the imaginary parts of the hopping energy. Besides, we also study the Lieb lattice with different intracell hopping and give the results of the geometrical phases of the model. In this case, the correlation phases will be always zero because the normalized coefficient is always a purely real number and the phase transition is vividly shown with the geometrical phases of the Majorana stars.

中文翻译：

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马约拉纳恒星表示下一维利布晶格的非厄米几何性质


非厄米哈密顿量的拓扑性质是一个热门话题，沿着这一研究方向的理论研究通常基于两级非厄米哈密顿量（或者等效地，自旋$1/2$非厄米哈密顿量） 。我们的动机是研究极化子凝聚态背景下具有平带的三级 Lieb 晶格模型（或等效的自旋 1$ 非厄米哈密顿量）的几何相。拓扑不变量是通过布里渊区中的绕数和布洛赫球中马约拉纳星的几何相位来计算的。此外，我们提供了一种直观的方法来研究具有较高自旋的拓扑相变，并且平带提供了定义布洛赫球上的拓扑相变的平台。根据马约拉纳星的运行轨迹，我们计算出马约拉纳星的几何相位。我们研究了具有复杂跳跃的Lieb晶格，发现当参数从平凡相变为拓扑相时，它们的相位发生跳跃。马约拉纳星的相关相位会随着跳跃能量虚部的增加而上升。此外，我们还研究了具有不同胞内跳变的Lieb晶格，并给出了模型的几何相位结果。在这种情况下，相关相位将始终为零，因为归一化系数始终是纯实数，并且相变通过马约拉纳星的几何相位生动地显示出来。