  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 $r^{-\alpha}$ 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 $\alpha>0.5$, 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.
