  In previous studies, the topological invariants of 1D non-Hermitian systems
have been defined in open boundary condition (OBC) to satisfy the bulk-boundary
correspondence. The extreme sensitivity of bulk energy spectra to boundary
conditions has been attributed to the breakdown of the conventional
bulk-boundary correspondence based on the topological invariants defined under
periodic boundary condition (PBC). Here we propose non-Hermitian many-body
polarization as a topological invariant for 1D non-Hermitian systems defined in
PBC, which satisfies the bulk-boundary correspondence. Employing many-body
methodology in the non-Hermitian Su-Schrieffer-Heeger model for fermions, we
show the absence of non-Hermitian skin effect due to the Pauli exclusion
principle and demonstrate the bulk-boundary correspondence using the invariant
defined under PBC. Moreover, we show that the bulk topological invariant is
quantized in the presence of chiral or generalized inversion symmetry. Our
study suggests the existence of generalized crystalline symmetries in
non-Hermitian systems, which give quantized topological invariants that capture
the symmetry-protected topology of non-Hermitian systems.
