Much has been learned about universal properties of the eigenstate entanglement entropy for one-dimensional lattice models, which is described by a Hermitian Hamiltonian, while much less has been understood for non-Hermitian systems. In the present work we study a non-Hermitian, noninteracting model of fermions which is invariant under combined $PT$ transformation. Our models show a phase transition from a $PT$ unbroken phase to broken phase as we tune the Hermiticity-breaking parameter. Entanglement entropy of such systems can be defined in two different ways, depending on whether we consider only right (or equivalently, only left) eigenstates or a combination of both left and right eigenstates which form a complete set of biorthonormal eigenstates. We demonstrate that the entanglement entropy of the ground state and also of the typical excited states shows some unique features in both of these phases of the system. Most strikingly, entanglement entropy obtained taking a combination of both left and right eigenstates shows an exponential divergence with system size at the transition point. While in the $PT$-unbroken phase, the entanglement entropy obtained from only the right (or equivalently, left) eigenstates shows identical behavior to an equivalent Hermitian system which is connected to the non-Hermitian system by a similarity transformation.

中文翻译：

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aPT 不变的非厄米系统中的本征态纠缠熵

关于一维晶格模型的本征态纠缠熵的普遍性质已经了解了很多，这是由 Hermitian Hamiltonian 描述的，而对非 Hermitian 系统的了解则少得多。在目前的工作中，我们研究了一个非厄米、非相互作用的费米子模型，它在组合条件下是不变的。$磷吨$转型。我们的模型显示了从$磷吨$当我们调整 Hermiticity-breaking 参数时，从不间断相到破裂相。这种系统的纠缠熵可以用两种不同的方式定义，这取决于我们是只考虑右（或等价地，只考虑左）本征态，还是左右本征态的组合，形成完整的双正交本征态集。我们证明了基态和典型激发态的纠缠熵在系统的这两个阶段中都显示出一些独特的特征。最引人注目的是，采用左右本征态的组合获得的纠缠熵在转换点与系统大小呈指数发散。而在$磷吨$- 不间断阶段，仅从右（或等效地，左）本征态获得的纠缠熵显示出与通过相似变换连接到非厄米系统的等效厄米系统相同的行为。