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 the steady state half-chain entanglement entropy and identify two distinct area law regions and an algebraic scaling region. Finite-size scaling analysis reveals critical scaling behavior of the half-chain entanglement entropy. 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 novel entanglement phases emerging from the interplay between the non-Hermitian skin effect and disorder-free localization.