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International Journal of Modern Physics B

Binding from Repulsive Interaction in a Degenerate Fermi Gas

Y. M. Malozovsky, J. D. Fan

Published Aug 10, 2003• 8 citations
Publisher

Abstract

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.

DOI
10.1142/s021797920302082x

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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 have examined the effective interaction of two fermions in terms of the T -matrix and shown that there are two different series of ladder diagrams in describing the effective interaction: (1) the exchange ladder diagrams giving rise to the T -matrix for the triplet state with poles for attractive interaction and (2) the Hartree ladder diagrams describing the multiple scattering on the Hartree potential and producing the T -matrix for the singlet state with poles for repulsive interaction. It is verified that a bound state of two fermions in the singlet state exists only in the presence of repulsive interaction between them. We further argue that the potential in the Schrodinger equation should be spin-dependent to describe both the singlet and triplet states.
We have examined the effective interaction of two fermions in terms of the T-matrix and shown that there are two different series of ladder diagrams in describing the effective interaction: (1) the exchange ladder diagrams giving rise to the T-matrix for the triplet state with poles for attractive interaction and (2) the Hartree ladder diagrams describing the multiple scattering on the Hartree potential and producing the T-matrix for the singlet state with poles for repulsive interaction. It is verified that a bound state of two fermions in the singlet state exists only in the presence of repulsive interaction between them. We further argue that the potential in the Schrodinger equation should be spin-dependent to describe both the singlet and triplet states.
The application of Brueckner's perturbation approach to interacting Fermion systems is prevented by the poles of the t-matrix which can be present for finite as well as infinite systems if the interaction is attractive. It is discussed that the singularity is removed if we take account of the contribution to the energy from diagrams involving two hole labels a number of times. These diagrams are not necessarily of ladder type and correspond to the possibility that a scattered pair of particles can go to the levels which they have left in the unperturbed state. Though the removal of poles makes one entertain a hope for the perturbation approach to finite systems, it is discussed that the same is not the case with infinite systems. It is pointed out that the difficulty in dealing with infinite systems is not caused by the pairs corresponding to the poles as is usually considered to be, but is due to pairs lying more closely to the Fermi surface.
The pairing instability induced by either a given attractive interaction or particle-hole excitations is examined by using the perturbational diagram approach. A graphical functional derivative method based on Ward's identity is developed 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 due to 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 scattering diagrams from this model allows one to evaluate the two particle vertex function and Tc by using the pole condition of the vertex function. It has been found that Tc can reach quite high values and gives rise to the bell shape. The application of the results to a layered 2D Fermi gas is considered.
We study the dilute-gas expansion for a two-dimensional Fermi system with arbitrary short-range repulsive interactions. In contrast with the three-dimensional case, we find an unusual pole in the vertex part in the particle-particle channel, for all center-of-mass momenta q2kF{\mathit{k}}_{\mathit{F}}kF​. We show that this represents an excitation consisting of bound hole pairs, which disperses down to zero energy at q=2kF{\mathit{k}}_{\mathit{F}}kF​. We study the effect of these bound states on the single-particle self-energy and find that the quasiparticles are well defined. Thus in the low-density regime, there is no breakdown of Fermi-liquid theory. In the Appendix we discuss the two-particle phase shift in the dilute Fermi gas, its connection with bound states, and the analogy with the potential scattering phase shift.
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.