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 SYK₂ 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 = 2α−1 / 2 and subsystem entanglement S_A ∝ L^(1−z)_A, 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 SYK₂ 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