We explore oscillatory behavior in a family of periodically driven spin chains which are subject to a weak measurement followed by postselection. We discover a transition to an oscillatory phase as the strength of the measurement is increased. By mapping these spin chains to free fermion models, we find that this transition is reflected in the opening of a gap in the imaginary direction. Interestingly, we find a robust, purely real, edge $\pi$ mode in the oscillatory phase. We establish a correspondence between the complex bulk spectrum and these edge modes. These oscillations are numerically found to be stable against interactions and disorder.

中文翻译：

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非酉 Floquet 系统中的鲁棒振荡和边缘模式

我们探索了一系列周期性驱动的自旋链中的振荡行为，这些自旋链受到弱测量和后选择的影响。我们发现随着测量强度的增加，会过渡到振荡阶段。通过将这些自旋链映射到自由费米子模型，我们发现这种转变反映在虚数方向上间隙的打开。有趣的是，我们发现了一个强大的、纯粹真实的边缘$\pi$模式处于振荡阶段。我们在复杂的体谱和这些边缘模式之间建立了对应关系。在数值上发现这些振荡对于相互作用和无序是稳定的。