  It is well-known that, generically, the one-dimensional interacting fermions
cannot be described in terms of the Fermi liquid. Instead, they present
different phenomenology, that of the Tomonaga-Luttinger liquid: the Landau
quasiparticles are ill-defined, and the fermion occupation number is continuous
at the Fermi energy. We demonstrate that suitable fine-tuning of the
interaction between fermions can stabilize a peculiar state of one-dimensional
matter, which is dissimilar to both the Tomonaga-Luttinger and Fermi liquids.
We propose to call this state a quasi-Fermi liquid. Technically speaking, such
liquid exists only when the fermion interaction is irrelevant (in the
renormalization group sense). The quasi-Fermi liquid exhibits the properties of
both the Tomonaga-Luttinger liquid and the Fermi liquid. Similar to the
Tomonaga-Luttinger liquid, no finite-momentum quasiparticles are supported by
the quasi-Fermi liquid; on the other hand, its fermion occupation number
demonstrates finite discontinuity at the Fermi energy, which is a hallmark
feature of the Fermi liquid. Possible realization of the quasi-Fermi liquid
with the help of cold atoms in an optical trap is discussed.
