  We demonstrate how Exceptional Points (EPs) naturally occur in the Luttinger
Liquid (LL) theory describing the low-energy excitations of a microscopic
lattice model with sublattice dependent electron-electron interaction. Upon
bosonization, this sublattice dependence directly translates to a non-standard
sine-Gordon-type term responsible for the non-Hermitian matrix structure of the
single-particle Green Function (GF). As the structure in the lifetime of
excitations does not commute with the underlying free Bloch Hamiltonian,
non-Hermitian topological properties of the single-particle GF emerge --
despite our Hermitian model Hamiltonian. Both finite temperature and a
non-trivial Luttinger parameter $K\neq 1$ are required for the formation of
EPs, and their topological stability in one spatial dimension is guaranteed by
the chiral symmetry of our model. In the presence of the aforementioned
sine-Gordon-term, we resort to leading order Perturbation Theory (PT) to
compute the single-particle GF. All qualitative findings derived within LL
theory are corroborated by comparison to both numerical simulations within the
conserving second Born approximation, and, for weak interactions and high
temperatures, by fermionic plain PT. In certain parameter regimes, quantitative
agreement can be reached by a suitable parameter choice in the effective
bosonized model.
