Coulomb Screening, Dielectric Response, and RPA Validity

Screening is the self-consistent rearrangement of charge induced by a perturbing potential. The random-phase approximation (RPA) is the geometric resummation of polarization bubbles that produces this rearrangement. It is exact for a specified noninteracting polarization inside that diagram class, but its accuracy for an interacting material depends on density, dimensionality, degeneracy, and the observable. Its controlled high-density correlation-energy use is demonstrated in Gell-Mann and Brueckner 1957, pp. 364–367.

Required background. Polarization, the Lindhard Function, and Particle–Hole Continua fixes the sign and normalization of $\Pi_0$ .

Helpful background. Sources, Linear Response, and Kubo Formulae supplies the response and analytic-continuation framework.

Self-consistent potential and dielectric function

Let $U_{\rm ext}$ couple to number density and let $v_q>0$ be the repulsive interaction. In the convention $\delta n=\Pi_0U_{\rm tot}$ , the induced Hartree potential is $v_q\delta n$ . Therefore

U_{\rm tot}=U_{\rm ext}+v_q\Pi_0U_{\rm tot}, \qquad U_{\rm tot}=\frac{U_{\rm ext}}{1-v_q\Pi_0}.

The RPA dielectric function, density response, and screened interaction are consequently

\epsilon_{\rm RPA}(q,\omega)=1-v_q\Pi_0^R(q,\omega),

\chi_{\rm RPA}^R=\frac{\Pi_0^R}{1-v_q\Pi_0^R}, \qquad W_{\mathrm{RPA}}^R=\frac{v_q}{1-v_q\Pi_0^R}.

Diagrammatically, $W=v+v\Pi_0v+v\Pi_0v\Pi_0v+\cdots$ . This identity also fixes every sign: in the static limit $\Pi_0<0$ , so repulsion makes the denominator larger than one and reduces the long-range potential.

Thomas–Fermi screening in three dimensions

For Coulomb interactions in a background dielectric medium,

v_q=\frac{4\pi e^2}{\epsilon_bq^2}.

At $T=0$ and $q\ll k_F$ , $\Pi_0(q,0)=-\nu_F+O(q^2)$ . Hence

\epsilon_{\mathrm{RPA}}(q,0)=1+\frac{k_{\mathrm{TF}}^2}{q^2}, \qquad k_{\mathrm{TF}}^2=\frac{4\pi e^2\nu_F}{\epsilon_b},

and

W_{\mathrm{RPA}}(q,0)=\frac{4\pi e^2}{\epsilon_b(q^2+k_{\mathrm{TF}}^2)}.

Fourier transformation gives a Yukawa potential $W(r)=e^2e^{-k_{\mathrm{TF}}r}/(\epsilon_b r)$ . The approximation captures the long-wavelength screening length; keeping the full Lindhard function also produces the $2k_F$ nonanalyticity and Friedel tail, which a constant $\Pi_0$ misses.

What RPA sums and what it omits

RPA sums direct ring diagrams built from bare or otherwise declared propagators and bare density vertices. It omits exchange diagrams, vertex corrections, and local-field factors. If dressed propagators are inserted without changing the vertex, conservation identities can be lost.

Controlled regimes include the high-density electron gas for leading correlation-energy logarithms, large flavor degeneracy with an appropriate interaction scaling, and long wavelengths where the Coulomb singularity enhances direct density fluctuations. These are regime statements, not a uniform error bound for every $q$ , $\omega$ , or observable. Short-range correlations, low density, and proximity to an instability generally require corrections.

For a neutral system, the $q=0$ Coulomb component is removed or cancelled by the uniform positive background. Failing to state that prescription creates a spurious divergent Hartree energy.

Analytic structure and measured loss

Retarded screening uses $\Pi_0^R(q,z)$ analytic for $\operatorname{Im}z>0$ . The energy-loss function

\mathcal L(q,\omega)=-\operatorname{Im}\epsilon^{-1}(q,\omega)

can contain both a collective pole and particle–hole background. A peak in $\mathcal L$ is not by itself proof of an isolated pole: one must continue the inverse response and distinguish a zero of $\epsilon$ from a maximum created by a continuum edge.

Common pitfalls

Using $1+v_q\Pi_0$ with a negative static polarization. That convention would antiscreen. Derive the denominator from the declared coupling rather than memorizing a sign.

Calling RPA generically controlled. Its leading terms are controlled only in specified limits. A small interaction parameter for one quantity need not control spectra near a continuum threshold.

Mixing background and vacuum Coulomb conventions. State whether $\epsilon_b$ and the neutralizing background have already been included in $v_q$ .

Exercises

Sum the bubble series

Starting from $W=v+v\Pi_0v+\cdots$ , derive the screened interaction for commuting scalar functions.

Solution

Factor out $v$ and recognize a geometric series: $W=v\sum_{n=0}^{\infty}(\Pi_0v)^n=v/(1-v\Pi_0)$ when the series converges. The retarded expression elsewhere follows by analytic continuation of this identity.

Check the screening sign

A positive static test potential is applied to an electron gas. Explain why $W<v$ at small momentum.