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Universal Entanglement Transitions of Free Fermions with Long-range Non-unitary Dynamics

Non-unitary evolution can give rise to novel steady states classified by their entanglement properties. In this work, we aim to understand its interplay with long-range hopping that decays with $r^{-α}$ in 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 $α>0.5$, we observe two critical phases with subvolume entanglement scaling: (i) $α>1.5$, a logarithmic phase with dynamical exponent $z=1$ and logarithmic subsystem entanglement, and (ii) $0.5<α<1.5$, a fractal phase with $z=\frac{2α-1}{2}$ 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.