Mermin's dielectric function and the f-sum rule

TL;DR

This paper critically examines Mermin's dielectric function, revealing a moment-closure problem, analyzing the conditions under which the f-sum rule holds, and discussing implications for fitting models to data.

Contribution

It identifies a moment-closure issue in Mermin's dielectric function and explores how different collision frequency models affect the f-sum rule's validity.

Findings

Collision frequencies scaling as ω violate the f-sum rule.

Constant, real, positive collision frequencies satisfy the f-sum rule.

Numerical evaluations show apparent violations due to slow convergence.

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 $\omega$ 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 $\omega_{\max}$ 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.