标题： $PT$对称的石墨烯在磁场下 显示英文标题

标题： $PT$-symmetric graphene under a magnetic field

摘要： 我们提出了一种在磁场下具有$PT$对称性变形的石墨烯紧束缚模型。 我们分析了哈密顿量在$K$和$K'$点附近的谱结构和本征向量，分别在$PT$对称和$PT$破坏区域中。 特别是我们展示了变形参数$V$的存在产生了一些有趣的结果，包括哈密顿量零能态的不对称性和本征向量集的不完备性。 我们还讨论了特征向量的双正交性，这在{结果是}中在$PT$对称和$PT$破坏区域有所不同。 显示英文摘要

摘要： We propose a $PT$-symmetrically deformed version of the graphene tight-binding model under a magnetic field. We analyze the structure of the spectra and the eigenvectors of the Hamiltonians around the $K$ and $K'$ points, both in the $PT$-symmetric and $PT$-broken regions. In particular we show that the presence of the deformation parameter $V$ produces several interesting consequences, including the asymmetry of the zero-energy states of the Hamiltonians and the breakdown of the completeness of the eigenvector sets. We also discuss the biorthogonality of the eigenvectors, which {turns out to be} different in the $PT$-symmetric and $PT$-broken regions.