It is well known that, generically, one-dimensional interacting fermions cannot be described in terms of a Fermi liquid. Instead, they present a different phenomenology, that of a 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 Tomonaga-Luttinger and Fermi liquids. We propose to call this state a quasi-Fermi liquid. Technically speaking, such a liquid exists only when the fermion interaction is irrelevant (in the renormalization group sense). The quasi-Fermi liquid exhibits the properties of both a Tomonaga-Luttinger liquid and a Fermi liquid. Similar to a Tomonaga-Luttinger liquid, no finite-momentum quasiparticles are supported by the quasi-Fermi liquid; on the other hand, its fermion occupation number demonstrates a finite discontinuity at the Fermi energy, which is a hallmark feature of a Fermi liquid. A possible realization of the quasi-Fermi liquid with the help of cold atoms in an optical trap is discussed.
We show that the traditional classification of quantum inpurity models based on thermodynamics is insufficient to probe the nature of their low-energy dynamics. We propose an analysis based on scattering theory, dividing Fermi liquids into regular Fermi liquids and singular Fermi liquids. In both cases electrons at the Fermi energy scatter elastically off the impurity, but in the case of regular Fermi liquids the scattering has analytical properties in the vicinity of the Fermi energy, while for singular Fermi liquids it does not, resulting in a breakdown of Nozi\`eres' Fermi-liquid picture and singular thermodynamic behavior. Using the Bethe ansatz and numerical renormalization group, we show that the ordinary Kondo model is a regular Fermi liquid whereas the underscreened Kondo model is a a singular Fermi liquid. Conventional regular Fermi liquid behavior is reestablished in an external magnetic field
H, but with a density of states which diverges as
1∕H. Our results may be relevant for the recently observed field-tuned quantum criticality in heavy electron materials.
We will consider the normal state of interacting Fermi particles without any longrange order. The normal state at low temperatures is called the Fermi liquid and is considered to be the system of free quasi-particles that is continuously connected with free Fermi gas. The concept of the Fermi liquid was introduced and developed by L. D. Landau. Landau's Fermi liquid theory, which concentrates rich contents into a simple theory, is a good example to solve many-body problems. In this chapter, we introduce the basis and main contents of the Fermi liquid. Principle of continuity The basis of the Fermi liquid theory is the principle of adiabatic continuity, which connects free Fermi gas with the Fermi liquid by introducing gradually an interaction among particles. There is a one-to-one correspondence between two states before and after the introduction of the interaction. States belonging to the same symmetry do not cross each other, and new states with the interaction can be represented by the old quantum number. Since the distribution of free Fermi gas is given by the Fermi distribution function n( k ) , that of the Fermi liquid is also written n ( k ). The state denoted by k , σ in the Fermi liquid is called the quasi-particle. For the system to be described by quasi-particles, the following condition is necessary. Let us consider the basis of the Fermi liquid theory following the explanation of the Landau theory given by Nozières and Anderson.