Xu Xingran, Liu Haodi, Zhang Zhidong, Liang Zhaoxin
Shenyang National Laboratory for Materials Science, Institute of Metal Research, Chinese Academy of Sciences, Shenyang, 110016, People's Republic of China.
School of Materials Science and Engineering, University of Science and Technology of China, Hefei, 230026, People's Republic of China.
J Phys Condens Matter. 2020 Jul 23;32(42). doi: 10.1088/1361-648X/ab9fd4.
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 on 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 calculate the geometrical phases of the model using non-Bloch factor under the Majorana's stellar representation. In this case, the correlation phases will always be zero because of the normalized coefficient is always a purely real number and the phase transition is vividly shown with the geometrical phases of the Majorana stars calculated by the mean values of the total phases of both right and the joint left eigenstates.
非厄米哈密顿量的拓扑性质是一个热门话题，沿着这条研究路线的理论研究通常基于两能级非厄米哈密顿量（或者等效地，一个自旋1/2非厄米哈密顿量）。我们受激发去研究在极化激元凝聚体背景下具有平带的三能级利布晶格模型（或者等效地，一个自旋1非厄米哈密顿量）的几何相位。拓扑不变量通过布里渊区中的缠绕数以及布洛赫球上马约拉纳星的几何相位来计算。此外，我们提供了一种直观的方法来研究具有更高自旋的拓扑相变，并且平带提供了一个在布洛赫球上定义拓扑相变的平台。根据马约拉纳星的轨迹，我们计算马约拉纳星的几何相位。我们研究具有复跳迁的利布晶格，发现当参数从平凡相变为拓扑相时它们的相位会有一个跳跃。马约拉纳星的关联相位将随着跳迁能量虚部的增加而上升。此外，我们还研究了具有不同格点内跳迁的利布晶格，并在马约拉纳星表示下使用非布洛赫因子计算该模型的几何相位。在这种情况下，由于归一化系数始终是一个纯实数，关联相位将始终为零，并且通过由右本征态和联合左本征态的总相位平均值计算出的马约拉纳星几何相位生动地展示了相变。
