The Cooper instability in an interacting Fermi gas is examined by using the perturbational diagram approach. A graphical derivative method based on Ward's identity is introduced to represent the contributions to the interparticle interactions from different channels. It is shown that pairing instability for the electrons excited from the Fermi sea may happen as a result of the presence of holes even when the original interaction is repulsive. Thus, in contrast to the Cooper instability caused by a given attractive interaction, the pairing instability can be induced by the particle - hole excitations. The application of the results to a layered two-dimensional Fermi gas is considered.
The Cooper instability in a Fermi gas is examined using the perturbative diagram approach. A graphical functional derivative technique based on Ward's identity is developed to obtain two-particle interactions and then to calculate the vertex part. The pairing instability for a given interaction, such as a phonon (plasmon, etc.) field, occurs in a quiescent Fermi sea, i.e. without exciting or involving background particles (holes), only if the interaction is attractive, as first proposed by Cooper and adopted in the BCS (Bardeen–Cooper–Schrieffer) theory. The consequence from this technique provides a way to evaluate the effect of the vertex corrections and responses of the Fermi gas in both charge and spin channels incorporating the backward scattering process. The significance of the methodology presented in the present work lies in the fact that it can both reproduce the known results, and, more importantly, be extended to investigate the intermediate or strong coupling case, such as nuclear interactions, where a neglect of vertex corrections may not be a good approximation.
By means of the Fermi-liquid approach, the effects of exchange and correlations are considered in a layered two-dimensional electron gas. A graphical derivative technique is introduced to obtain the contributions to the interparticle interactions from different channels. It is shown that the interaction in the particle - hole channel can induce the metal - insulator phase transition when the Fermi momentum is comparable with the inverse interlayer spacing, . It is also shown that the Cooper-like scattering processes give rise to the scaling of the Fermi liquid response as Im , which leads to the linear temperature dependence of the quasiparticle damping. The possibility of superconductivity in the region between the insulator and normal metal phases, accounting for the particle - hole excitations, is discussed. The effect of the spin fluctuations on the superconductivity due to the particle - hole excitations is also considered.
The binding of two fermions in a degenerate Fermi gas is examined in terms of the T-matrix. We deal with effective T-matrix of two fermions in the degenerate Fermi gas in terms of the scattering amplitude. This makes approach valid for the retarded as well as non-retarded interactions. We show that for the interaction of a pair of fermions in a quiescent Fermi sea there are two different series of the ladder diagrams-Exchange and Hartree diagrams, describing a bound state and producing two different T-scattering matrices. The exchange series of the ladder diagrams corresponds to the T-matrix in the triplet-state. This T-matrix contains the pole solution corresponding to the bound state for the attractive interaction only. In contrast, the Hartree diagrams lead to the T-matrix of a pair of fermions in the singlet-state, which have a pole solution corresponding to the bound state of a pair of fermions for the repulsive interaction. Thus, we are led to assert that the bound state of a pair of fermions in the singlet-state in the quiescent Fermi sea can form for the repulsive interaction only and there is no bound state in the singlet-state from the attractive interaction.
We consider possible superconducting instabilities in a two-dimensional Fermi system with short-ranged repulsive interactions between electrons. The possibility of an unusual superconducting paring due to the Kohn-Luttinger mechanism is examined. The quasiparticle scattering amplitude is shown to possess an attractive harmonic in second-order perturbation theory for finite values of the energy transfer. The corresponding singularity in the pairing vertex leads to a superconducting pairing of the electron excitations with finite energies. We identify the energy transfer in the Cooper channel as the binding energy of the excited pair. At low enough temperatures, the Fermi system is a mixture of normal electron excitations and fluctuating d-wave Cooper pairs possessing a finite gap.
Interacting Fermi gas provides an ideal model system to understand unconventional pairing and intertwined orders relevant to a large class of quantum materials. Rydberg-dressed Fermi gas is a recent experimental system where the sign, strength, and range of the interaction can be controlled. The interaction in momentum space has a negative minimum at
qc inversely proportional to the characteristic length scale in real space, the soft-core radius
rc. We show theoretically that single-component (spinless) Rydberg-dressed Fermi gas in two dimensions has a rich phase diagram with superfluid and density wave orders due to the interplay of the Fermi momentum
pF, interaction range
rc, and interaction strength
u0. For repulsive bare interactions
u0>0, the dominant instability is a
f-wave superfluid for ${p}_{F}{r}_{c}\ensuremath{\lesssim}2$ and a density wave for ${p}_{F}{r}_{c}\ensuremath{\gtrsim}4$. The
f-wave pairing in this repulsive Fermi gas is reminiscent of the conventional Kohn-Luttinger mechanism but has a much higher
Tc. For attractive bare interactions
u0<0, the leading instability is
p-wave pairing. The phase diagram is obtained from a functional renormalization group that treats all competing many-body instabilities in the particle-particle and particle-hole channels on an equal footing.