Linking critical temperature with electron localization for cavity-enhanced superconductivity

Predicting superconducting properties from first principles-especially in non-equilibrium conditions-is computationally intensive. Here, we propose a more efficient approach by using the electron localization function (ELF) as a proxy for estimating the superconducting critical temperature T _C. Through first-principles calculations, we investigate how coupling conventional superconductors to an optical cavity-without external driving-modifies their phonon properties and electron-phonon interactions via vacuum fluctuations alone. We focus on three representative materials: lead (Pb), niobium (Nb), and magnesium diboride (MgB₂). Our methodology combines Density Functional Theory (DFT), Density Functional Perturbation Theory (DFPT), Quantum Electrodynamical Density Functional Theory (QEDFT), and Wannier-based electron-phonon coupling to solve the Eliashberg equations for T _C. For the materials studied here, our results indicate that the ELF captures some trends in the superconducting behavior under light-matter coupling, suggesting it may serve as a low-cost descriptor to guide the screening or design of superconductors in equilibrium and cavity-modified regimes.