  We present numerical evidence for a paradigm in one-dimensional interacting
fermion systems, whose phenomenology has traits of both Luttinger liquids and
Fermi liquids. This state, dubbed a quasi-Fermi liquid, possesses a
discontinuity in its fermion occupation number at the Fermi momentum. The
excitation spectrum presents particlelike quasiparticles and absence of
holelike quasiparticles, giving rise instead to edge singularities. Such a
state is realized in a one-dimensional spinless fermion lattice Hamiltonian by
fine-tuning the interactions to a regime where they become irrelevant in the
renormalization group sense. We show, using uniform infinite matrix products
states and finite-entanglement scaling analysis, that the system ground state
is characterized by a Luttinger parameter $K = 1$ and a discontinuous jump in
the fermion occupation number. We support the characterization with
calculations of the spectral function that show a particle-hole asymmetry
reflected in the existence of well-defined Landau quasiparticles above the
Fermi level and edge singularities without the associated quasiparticles below.
These results indicate that the quasi-Fermi liquid paradigm can be realized
beyond the low-energy perturbative realm.
