The exactly solvable Luttinger model for a system of one-dimensional, massless, two-component fermions interacting via general two-body forces is used as a test case for a systematically generated sequence of decoupling approximations commonly referred to as random-phase approximations (RPA) of which the simple RPA is the lowest. It is found, irrespective of both the nature and strength of the interaction, that the exact (collective) spectrum, as well as the exact correlation energy, can already be obtained in the simple RPA, and furthermore, that the next higher RPA gives a vanishing correction to these quantities. It is also shown, quite in contradistinction to the exact result, that the free-gas momentum distribution ${\overline{n}}_{k}$ persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of ${\overline{n}}_{k}$ . Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.The exactly solvable Luttinger model for a system of one-dimensional, massless, two-component fermions interacting via general two-body forces is used as a test case for a systematically generated sequence of decoupling approximations commonly referred to as random-phase approximations (RPA) of which the simple RPA is the lowest. It is found, irrespective of both the nature and strength of the interaction, that the exact (collective) spectrum, as well as the exact correlation energy, can already be obtained in the simple RPA, and furthermore, that the next higher RPA gives a vanishing correction to these quantities. It is also shown, quite in contradistinction to the exact result, that the free-gas momentum distribution ${\overline{n}}_{k}$ persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of ${\overline{n}}_{k}$. Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.

Physical ReviewApr 5, 19682 citations

Collective excitations in fermi systems

Viktor Mikhaĭlovich Galit︠s︡kiĭ

PhysicaDec 1, 19603 citations

Collective Excitations of Trapped Bose–Fermi Mixture of Atomic Gases in RPA

Takaaki Sogo, Takahiko Miyakawa, T. Suzuki +1 more

Collective excitations of a trapped Bose–Fermi mixture of atomic gases are studied in a random-phase approximation. Density profiles in the ground states are solved in the mean field approximation for various strengths of Bose–Fermi interaction. Monopole strength distributions and the dynamical structure factors are investigated for two types of the Bose–Fermi mixture. It is found in the mixture with the large excess of bosons the fermionic strength is concentrated in a single collective state for strong Bose–Fermi interactions. The dynamical structure factors for these collective states show that fermions are correlated through the Bose condensate. Time dependent behavior of the system after an external impulse is also studied.Collective excitations of a trapped Bose–Fermi mixture of atomic gases are studied in a random-phase approximation. Density profiles in the ground states are solved in the mean field approximation for various strengths of Bose–Fermi interaction. Monopole strength distributions and the dynamical structure factors are investigated for two types of the Bose–Fermi mixture. It is found in the mixture with the large excess of bosons the fermionic strength is concentrated in a single collective state for strong Bose–Fermi interactions. The dynamical structure factors for these collective states show that fermions are correlated through the Bose condensate. Time dependent behavior of the system after an external impulse is also studied.

Journal of the Physical Society of JapanJan 1, 20030 citations

The RPA of renormalized fermions

G.G. Dussel, R.P.J. Perazzo, María Elena Spina

Nuclear Physics AJan 1, 19810 citations

Single-particle and collective excitations in a solvable many-fermion system

R. W. Richardson

Annals of PhysicsJan 1, 19712 citations

{\overline{n}}_{k}

persists in the second RPA as well as in the first, thereby eliminating the higher RPA as a possible approach to the calculation of

{\overline{n}}_{k}

. Thus, in the context of this particular model, although the first few orders of RPA are highly effective for calculating the excitation spectrum and correlation energy, they are sensibly powerless in their ability to predict any variation in structure beyond that of the free-gas distribution, a result which most probably holds as well in the realistic three-dimensional case.