Reply to: Re-examining magnetic tuning of Casimir forces

replying to M. Moazzami Gudarzi and S. H. Aboutalebi Nature Physics https://doi.org/10.1038/s41567-025-02845-5 (2025)

In their Matters Arising, Moazzami Gudarzi and Aboutalebi¹ first highlight the inaccurate dielectric functions of water and Fe₃O₄ used in our Lifshitz theory calculations¹. Upon careful re-examination, we discovered that our representation of \(\varepsilon_{{{{\rm{Fe}}_{3}}}{{\rm{O}}}_{4}}(i\xi)\) (where \(\varepsilon_{{{{\rm{Fe}}_{3}}}{{\rm{O}}}_{4}}\) is the dielectric permittivity of Fe₃O₄, i is the imaginary unit and ξ is the Matsubara frequency) was based on a mistyped equation (equation (9)) from ref. ²:

$${I}_{3}\left(\xi\,\right)=\frac{C{\varOmega }_{2}^{2}}{{\xi }^{\,2}}\left[1+\frac{{\varOmega }_{2}}{\xi }\arctan \frac{{\varOmega }_{2}}{\xi }-\frac{\uppi }{2}\right],$$

where I₃ is a part of Kramers–Kronig relation, C = 1.58 and Ω₂ = 1.8 × 10⁶ rad/s. The correct equation should read:

$${I}_{3}\left(\xi\,\right)=\frac{C{\varOmega }_{2}^{2}}{{\xi }^{2}}\left[1+\frac{{\varOmega }_{2}}{\xi }\arctan \frac{{\varOmega }_{2}}{\xi }-\frac{{{{\uppi }}\varOmega }_{2}}{2\xi }\right].$$