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.
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