We present a numerical study of Anderson localization in disordered non-Hermitian lattice models with flat bands. Specifically we consider one-dimensional stub and two-dimensional kagome lattices that have a random scalar potential and a uniform imaginary vector potential and calculate the spectra of the complex energy, the participation ratio, and the winding number as a function of the strength of the imaginary vector potential, $h$. The flat-band states are found to show a double transition from localized to delocalized and back to localized states with $h$, in contrast to the dispersive-band states going through a single delocalization transition. When $h$ is sufficiently small, all flat-band states are localized. As $h$ increases above a certain critical value $h_1$, some pair of flat-band states become delocalized. The participation ratio associated with them increases substantially and their winding numbers become nonzero. As $h$ increases further, more and more flat-band states get delocalized until the fraction of the delocalized states reaches a maximum. For larger $h$ values, a re-entrant localization takes place and, at another critical value $h_2$, all flat-band states return to compact localized states with very small participation ratios and zero winding numbers. This re-entrant localization transition, which is due to the interplay among disorder, non-Hermiticity, and flat band, is a universal phenomenon occurring in all models having an imaginary vector potential and a flat band simultaneously. We explore the spatial characteristics of the flat-band states by calculating the local density distribution.

中文翻译：

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平带无序晶格模型中虚矢量势诱导平带态的离域和重入定位

我们提出了在具有平坦带的无序非厄米晶格模型中安德森定位的数值研究。具体来说，我们考虑具有随机标量势和均匀虚矢量势的一维短截线和二维 kagome 晶格，并计算复能谱、参与比和绕组数作为强度的函数虚向量势，$h$。与通过单个离域跃迁的色散带状态相比，发现平带状态显示出从局部到离域并返回到局部状态的双重跃迁。当 $h$ 足够小时，所有的平带状态都是局部化的。随着 $h$ 增加到某个临界值 $h_1$ 以上，一些平带状态对变得离域。与它们相关的参与率显着增加，并且它们的绕组数变为非零。随着 $h$ 的进一步增加，越来越多的平带状态变得离域，直到离域状态的比例达到最大值。对于较大的 $h$ 值，会发生可重入局部化，并且在另一个临界值 $h_2$ 处，所有平带状态都会返回到具有非常小的参与比和零绕组数的紧凑局部化状态。这种重入定位转换是由于无序、非 Hermicity 和平带之间的相互作用而导致的，这是一种普遍现象，发生在同时具有虚矢量势和平带的所有模型中。我们通过计算局部密度分布来探索平带状态的空间特征。