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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 withrin non-Hermitian free-fermion systems. We first study two solvable Brownian models with long-range non-unitary dynamics: a large-NSYK2chain and a single-flavor fermion chain and we show that they share the same phase diagram. When0.5, we observe two critical phases with subvolume entanglement scaling: (i)1.5, a logarithmic phase with dynamical exponentz=1and logarithmic subsystem entanglement, and (ii)0.51.5, a fractal phase withz=212and subsystem entanglementSALA1z, whereLAis the length of the subsystemA. These two phases cannot be distinguished by the purification dynamics, in which the entropy always decays asL/T. We then confirm that the results are also valid for the static SYK2chain, 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.
Zhang et al. (2022) studied this question.