Mermin's dielectric function and the f-sum rule

Abstract

Mermin's dielectric function [N.D. Mermin, Phys. Rev. B 1, 2362 (1970)] is widely assumed to satisfy the f-sum rule because he constrains his ansatz with the continuity equation. However, we identify a moment-closure problem in Mermin's use of the continuity equation. Further, we show that the Mermin's model can be derived without invoking continuity. We describe how other approaches such as the ``completed Mermin'' model of Chuna and Murillo [Phys. Rev. E 111, 035206 (2025)] remedy this closure issue. We then inspect the f-sum rule for both the original and completed Mermin models and find for the Mermin ansatz that collision frequencies scaling as $ω$ must violate the f-sum rule, whereas constant, real, positive collision frequencies will satisfy it, with the caveat that, in practice, convergence with respect to the upper integration limit $ω_{\max}$ is sufficiently slow that finite-domain numerical evaluations exhibit apparent violations, regardless of wavenumber $q$. We also find that collision frequencies with constant imaginary components cause f-sum rule violations. We conclude that if Mermin's model is fit to data via optimizing its collision frequency, then the f-sum rule is not inherently satisfied; constraints, though broad, are needed in order to assume the f-sum rule is satisfied. Further, if the f-sum rule is theoretically satisfied, but violations still appear, then these deviations ought to be included in the error estimates.

Figures (4)

• Figure 1: Plot of the DSF obtained at wavenumber $q=0.88 q_F$, $\Theta=0.0855$, $r_s=2.07$, and $\nu = 0.5 \, \omega_p$ for the quantum Fermi-Dirac distribution; these condition are reflective of the large $\omega$ limit for Aluminum at 1 eV, see Hentschel et al.hentschel_PoP_2023. The percentage to which the f-sum rule is satisfied at a given $\omega_{\max}$ is indicated by colored text adjacent to the vertical line. The color and order matches that of the legend.
• Figure 2: Plot of the relative error in the frequency sum (f-sum) rule \ref{['eq:fsum_dielectric']} of the Mermin (green) and completed Mermin (CM, red) models computed using \ref{['eq:fsum_numeric']} with grid spacing $\Delta \omega = \omega_p/100$ and variable integration bounds. These computations are carried out at $r_s = 2.07$, $\Theta=0.0855$, $\nu = 0.5 \, \omega_p$, and four different wavenumbers $q/q_F$. For comparison, we plot the long wavelength limit of Mermin's f-sum rule convergence \ref{['eq:Merminfsumconvergence']} as a black line.
• Figure 3: Plot of the ratio of the inverse dielectric function's first moment, evaluated via \ref{['eq:fsum_numeric']}, to the expected value of $-\pi$ for the RPA (blue), Mermin (green) and completed Mermin (CM, red) models across wavenumber $q$ at $r_s = 2.07$, $\Theta=0.0855$, and $\nu= 0.5 \, \omega_p$ using both the classical Maxwellian equilibrium distribution and the quantum Fermi-Dirac distribution. Notice that the classical computations have been displaced by $\delta=0.04$, so the relative error is equivalent to the Lindhard case.
• Figure 4: Heatmaps indicating the ratio of the inverse dielectric function's first moment, computed via \ref{['eq:fsum_numeric']}, to the expected value of $-\pi$. Evaluations conducted at $r_s = 2.07$, $\Theta=0.0855$, $q=0.88 q_F$, and $\omega_{\max} = 40 \, \omega_p$ with Fermi-Dirac statistics with different constant collision frequencies $\nu(\omega) = \nu_0$, where $\nu(-\omega) = \nu_0^*$ so that its Fourier transform is real. Red arrows start from the RPA approximation $\nu = 0 + i 0$ and move outward. The rightward arrow indicates the effect of increasing the peak width. The upward arrow indicates the effect of shifting the peak to larger $\omega$, and the leftward arrow indicates the effect of shifting the peak to smaller frequencies. Notice the color bars differ.