Fate of non-Hermitian free fermions with Wannier-Stark ladder

The Wannier-Stark localization dynamically alters the entanglement behavior of non-Hermitian free fermions. Utilizing the single-particle correlation matrix technique, we analyze the effective Hamiltonian of these fermions with a Wannier-Stark ladder. Under open boundary conditions, we observe steady-state half-chain entanglement entropy (EE) and identify two distinct area-law regions and an algebraic scaling region. Finite-sized scaling analysis reveals critical scaling behavior of the half-chain EE. Notably, the system demonstrates unique entanglement characteristics under periodic boundary conditions, which diverge from the $(1+1)d$ conformal field theory predictions for Anderson localization. Our findings highlight entanglement phases emerging from the interplay between the non-Hermitian skin effect and disorder-free localization.