  Recently, the steady states of non-unitary free fermion dynamics are found to
exhibit novel critical phases with power-law squared correlations and a
logarithmic subsystem entanglement. In this work, we theoretically understand
the underlying physics by constructing solvable static/Brownian quadratic
Sachdev-Ye-Kitaev chains with non-Hermitian dynamics. We find the action of the
replicated system generally shows (one or infinite copies of) $O(2)\times O(2)$
symmetries, which is broken to $O(2)$ by the saddle-point solution. This leads
to an emergent conformal field theory of the Goldstone modes. We derive their
effective action and obtain the universal critical behaviors of squared
correlators. Furthermore, the entanglement entropy of a subsystem $A$ with
length $L_A$ corresponds to the energy of the half-vortex pair $S\sim \rho_s
\log L_A$, where $\rho_s$ is the stiffness of the Goldstone mode. We also
discuss special limits where more than one Goldstone mode exists and comment on
interaction effects.
