  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 the 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 total stiffness of the Goldstone
modes. We also discuss special limits with more than one branch of Goldstone
modes and comment on interaction effects.
