Non-Hermitian topology driven by an identity term: An exactly solvable paradigm

An identity term in the Hamiltonian is conventionally regarded as spectrally inert-it shifts energies but does not alter eigenstate topology. We show that under non-Hermitian skin pumping, this paradigm fails: a momentum-dependent identity term actively deforms the generalized Brillouin zone, thereby challenging established topological criteria that rely on fixed complex contours. Here, by introducing spin-orbit coupling into a Hatano-Nelson chain, we present an exact analytical solution for the entire non-Hermitian eigensystem under open boundary conditions. Our solution reveals how inter-cell spin-orbit coupling, synergizing with this non-trivial identity term, induces topological edge states and robust zero modes in the complete absence of chiral symmetry. This work establishes an exactly solvable paradigm for non-Hermitian topology beyond symmetry protection, and provides a rigorous benchmark for testing topological invariants in systems with momentum-dependent identity terms.