  In this paper, we study the conditions under which on-site dissipations can
induce non-Hermitian skin modes in non-Hermitian systems. When the original
Hermitian Hamiltonian has spinless time-reversal symmetry, it is impossible to
have skin modes; on the other hand, if the Hermitian Hamiltonian has spinful
time-reversal symmetry, skin modes can be induced by on-site dissipations under
certain circumstance. As a concrete example, we employ the Rice-Mele model to
illustrate our results. Furthermore, we predict that the skin modes can be
detected by the chiral tunneling effect, that is, the tunneling favors the
direction where the skin modes are localized. Our work reveals a no-go theorem
for the emergence of skin modes, and paves the way for searching for quantum
systems with skin modes and studying their novel physical responses.
