Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Log inSign up
Challenges
Datasets
Workspaces
Papers
Discussions
Leaderboard
Blog
Job Board
Q3AS

© 2026 Aqora Quantum S.A.S.

TermsPrivacyLegal Notice

arXiv:1111.5337

Lindhard function of a d-dimensional Fermi gas

Bogdan Mihaila

Published Nov 2, 2011• 16 citations
arXivHTMLPDF

Abstract

We review in detail the derivation of the dielectric response function of a noninteracting system of spin-1/2 fermions in the random-phase approximation. Results for the response function of a Fermi gas in one, two and three dimensions can be obtained in closed form, and represent the baseline for developing a pedagogical understanding of the effect of correlations on the response functions in interacting systems of fermions.

DOI
10.48550/arxiv.1111.5337

Related papers

Properties of an interacting one-dimensional fermion system

W. I. Friesen

For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.

•Jan 1, 2010•0 citations

Density response of one-, two-, and three-dimensional fermion systems with repulsive δ -function interaction

Seido Nagano, K. S. Singwi

A closed-form formula for the density-density response function of a system of fermions interacting via a repulsive $\ensuremath{\delta}$-function potential in one, two, and three dimensions is given in an approximation in which the particle-hole interaction in the polarization bubble is considered to all orders. The dynamic and static structure factors and the local-field factor have been calculated.

Physical review. B, Condensed matter•Dec 1, 1983•3 citations

Wigner-Kirkwoodħexpansion of the linear-response function for inhomogeneous Fermi systems

Guy Chanfray, Peter Schuck

For the treatment of inhomogeneous Fermi systems we derive a semiclassical expansion of the linear-response function in powers of \ensuremath{\Elzxh} (up to ${\ensuremath{\Elzxh}}^{2}$) for the interacting (random-phase approximation) and noninteracting (Lindhardt function) case. We find that the series is rapidly converging and that the semiclassical result compares very well with exact model calculations for cases where the response does not exhibit narrow resonances (e.g., the quasielastic peak region); otherwise, it still represents the average value.

Physical Review A•Nov 1, 1988•8 citations

The Random Phase Approximation for Interacting Fermi Gases in the Mean-Field Regime

Martin Ravn Christiansen, Christian Hainzl, Phan Thành Nam

Abstract We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to $N^{-\frac {1}{3}}$ . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.

arXiv:2106.11161•Forum of Mathematics Pi•Jan 1, 2023•17 citations

Effective pair potentials for the noninteracting spin-1/2 Fermi gas

F. A. Stevens, M. A. Pokrant

The N-particle configurational probability density P(N) of a noninteracting Fermi gas may be approximated as P(N) ≃ Πi<je−u(rij) / ∫d 3NrΠi<je −u(rij), where u(r) is an effective pair potential. Our reason for considering these potentials is that they have been successfully used in computing properties of interacting Fermi systems. We define the best effective potential to be the one whose corresponding two-body distribution function is closest to the exact two-body distribution function. A method for calculating the best u(r) is not known. Lado has calculated u(r) by using the Born-Green-Yvon integrodifferential equation relating u(r) to the two- and three-body distribution functions. Effective potentials may also be calculated using the hypernetted chain and Percus-Yevick integral equations. It is found that potentials computed by the above three methods are in fairly close agreement. The principal purpose of this paper is to present tables of effective potentials approximating both the spin-summed system and parallel spin subsystem probability densities for a system of noninteracting spin-1/2 fermions.

The Journal of Chemical Physics•Nov 1, 1974•7 citations

Classical calculation of correlation functions in a one-dimensional Fermi gas

J. N. Fields, Richard A. Klemm

The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.

Annals of the Israel Physical Society•Jan 1, 1978•0 citations
We review in detail the derivation of the dielectric response function of a noninteracting system of spin-1/2 fermions in the random-phase approximation. Results for the response function of a Fermi gas in one, two and three dimensions can be obtained in closed form, and represent the baseline for developing a pedagogical understanding of the effect of correlations on the response functions in interacting systems of fermions.
For nearly a decade, quasi-one-dimensional conductors have been the subject of intensive study. Theoretically, much attention has been devoted to the development of one-dimensional Fermi gas models, some which may be solved exactly, and to the calculation of their response functions. After a review of this theory, a different approach is adopted in the investigation of two models. The dielectric response theory of the three-dimensional Coulomb gas has been applied to an anisotropic system in which the particles interact with an effective one-dimensional long-range potential. Within the framework of the approximation of Singwi, Tosi, Land, and Sjolander, the dielectric properties of the model are examined in order to determine the conditions under which it is unstable with respect to formation of a charge density wave state. It is found that the positive neutralizing background must be polarizable in order for such an instability to occur. The same approximation method, when applied to a one-dimensional fermion gas with a ʃ-function interaction may be compared with the exact solution of Yang. This solution, which exists in the form of coupled integral equations, has been calculated numerically, and, as predicted by the Lieb-Mattis theorem, the ground state is found to be non-magnetic. The approximation of Singwi et al. proves to give better correlation energies than other inexact methods, particularly at higher densities.
For the treatment of inhomogeneous Fermi systems we derive a semiclassical expansion of the linear-response function in powers of \ensuremath{\Elzxh} (up to ${\ensuremath{\Elzxh}}^{2}$) for the interacting (random-phase approximation) and noninteracting (Lindhardt function) case. We find that the series is rapidly converging and that the semiclassical result compares very well with exact model calculations for cases where the response does not exhibit narrow resonances (e.g., the quasielastic peak region); otherwise, it still represents the average value.
Abstract We present a general approach to justify the random phase approximation for the homogeneous Fermi gas in three dimensions in the mean-field scaling regime. We consider a system of N fermions on a torus, interacting via a two-body repulsive potential proportional to N−13N^{-\frac {1}{3}}N−31​ . In the limit $N\rightarrow \infty $ , we derive the exact leading order of the correlation energy and the bosonic elementary excitations of the system, which are consistent with the prediction of the random phase approximation in the physics literature.
The N-particle configurational probability density P(N) of a noninteracting Fermi gas may be approximated as P(N) ≃ Πi<je−u(rij) / ∫d 3NrΠi<je −u(rij), where u(r) is an effective pair potential. Our reason for considering these potentials is that they have been successfully used in computing properties of interacting Fermi systems. We define the best effective potential to be the one whose corresponding two-body distribution function is closest to the exact two-body distribution function. A method for calculating the best u(r) is not known. Lado has calculated u(r) by using the Born-Green-Yvon integrodifferential equation relating u(r) to the two- and three-body distribution functions. Effective potentials may also be calculated using the hypernetted chain and Percus-Yevick integral equations. It is found that potentials computed by the above three methods are in fairly close agreement. The principal purpose of this paper is to present tables of effective potentials approximating both the spin-summed system and parallel spin subsystem probability densities for a system of noninteracting spin-1/2 fermions.
The response functions of the one-dimensional Fermi gas with backscattering are studied using an equivalence of the spin-density part of the Fermi Hamiltonian to the sine-Gordon Hamiltonian. The singlet spin-density response function is evaluated in functional-integral form, using a transfer-matrix technique. For low temperatures and large distances, this function is found to be unity. The triplet response function is found, in a harmonic approximation, to be zero. These results support the proposal by Luther and Emery that the singular behavior of the Fermi-gas response functions is determined solely by the charge-density modes. Also supported is the rather intuitive picture, based on a mapping to the two-dimensional Coulomb gas, of Chui and Lee.