Extended Data Fig. 5: Cavity-induced renormalization of the free energy of the metallic phase.

a. Free energy model setting. Upper panels: coplanar cavity with a thin slab of matter (thickness d) inside a cavity of length \(L\). Lower panels: Sketch of the cavity modes dispersion and of the absorption solid band (green shaded region centered at \({\omega }_{{diss}}\)). As \(L\) is increased, modes are pulled inside and below the absorption band of the solid. The cavity fundamental mode is indicated with \({\omega }_{c}(L).\) b. Dielectric loss spectrum \(\alpha {\prime\prime} (\omega \)) (\(\Omega \) = 15 GHz, \(\gamma \) = 20 GHz) employed for the calculations. The spectrum has been normalized by the static contribution to the polarizability \(\alpha (0)\). c. Renormalization of the metallic free energy \(\varDelta {F}_{m}\) as a function of the cavity frequency for different temperatures. The cavity frequencies \({\omega }_{c}\) are normalized by \(\Omega \) = 15 GHz. d. Renormalization of the metallic free energy \(\varDelta {F}_{m}\) as a function of the temperature for different cavity frequencies above and below resonance \({\omega }_{c}=\Omega \).