Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand the effect of long-range hopping that decays with $r^{-\alpha}$ in non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-$N$ SYK$_2$ chain and a single-flavor fermion chain and we show that they share the same phase diagram. When $\alpha>0.5$, we observe two critical phases with subvolume entanglement scaling: (i) $\alpha>1.5$, a logarithmic phase with dynamical exponent $z=1$ and logarithmic subsystem entanglement, and (ii) $0.5 < \alpha < 1.5$, a fractal phase with $z=\frac{2\alpha-1}{2}$ and subsystem entanglement $S_A\propto L_A^{1-z}$, where $L_A$ is the length of the subsystem $A$. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays as $L/T$. We then confirm that the results are also valid for the static SYK$_2$ chain, indicating the phase diagram is universal for general free-fermion systems. We also discuss phase diagrams in higher dimensions and the implication in measurement-induced phase transitions.

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具有长程非酉动力学的自由费米子的普遍纠缠跃迁

非统一演化可以产生按纠缠特性分类的新稳态。在这项工作中，我们旨在了解在非厄米特自由费米子系统中随 $r^{-\alpha}$ 衰减的远程跳跃的影响。我们首先研究了两个具有长程非单一动力学的可解布朗模型：一个大的$N$ SYK$_2$ 链和一个单味费米子链，我们证明它们共享相同的相图。当 $\alpha>0.5$ 时，我们观察到两个具有子体积纠缠缩放的关键阶段：（i）$\alpha>1.5$，一个动态指数 $z=1$ 和对数子系统纠缠的对数阶段，以及（ii）$0.5 < \alpha < 1.5$，一个具有 $z=\frac{2\alpha-1}{2}$ 和子系统纠缠 $S_A\propto L_A^{1-z}$ 的分形相位，其中 $L_A$ 是子系统$A$。这两个阶段无法通过纯化动力学来区分，其中熵总是衰减为 $L/T$。然后我们确认结果对静态 SYK$_2$ 链也有效，表明相图对于一般自由费米子系统是通用的。我们还讨论了更高维度的相图以及测量引起的相变的含义。