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It is well-known that, generically, the one-dimensional interacting fermionscannot be described in terms of the Fermi liquid. Instead, they presentdifferent phenomenology, that of the Tomonaga-Luttinger liquid: the Landauquasiparticles are ill-defined, and the fermion occupation number is continuousat the Fermi energy. We demonstrate that suitable fine-tuning of theinteraction between fermions can stabilize a peculiar state of one-dimensionalmatter, which is dissimilar to both the Tomonaga-Luttinger and Fermi liquids.We propose to call this state a quasi-Fermi liquid. Technically speaking, suchliquid exists only when the fermion interaction is irrelevant (in therenormalization group sense). The quasi-Fermi liquid exhibits the properties ofboth the Tomonaga-Luttinger liquid and the Fermi liquid. Similar to theTomonaga-Luttinger liquid, no finite-momentum quasiparticles are supported bythe quasi-Fermi liquid; on the other hand, its fermion occupation numberdemonstrates finite discontinuity at the Fermi energy, which is a hallmarkfeature of the Fermi liquid. Possible realization of the quasi-Fermi liquidwith the help of cold atoms in an optical trap is discussed.

One-Dimensional Fermions with neither Luttinger-Liquid nor Fermi-Liquid Behavior