Quantum criticality in the nonunitary dynamics of $(2+1)$-dimensional free fermions

We explore the nonunitary dynamics of $(2+1)$-dimensional free fermions and show that the obtained steady state is critical regardless the strength of the nonunitary evolution. Numerical results indicate that the entanglement entropy has a logarithmic violation of the area-law and the mutual information between two distant regions decays as a power-law function. In particular, we provide an interpretation of these scaling behaviors in terms of a simple quasiparticle pair picture. In addition, we study the dynamics of the correlation function and demonstrate that this system has dynamical exponent $z=1$. We further demonstrate the dynamics of the correlation function can be well captured by a classical nonlinear master equation. Our method opens a door to a vast number of nonunitary random dynamics in free fermions and can be generalized to any dimensions.