The Universal Gap-to-Critical Temperature Ratio in Superconductors: a Statistical Mechanical Perspective

We propose a statistical mechanical framework to unify the observed relationship between the superconducting energy gap $\mathrm{\Delta }$, the pseudogap ${\mathrm{\Delta }}^{\ast }$, and the critical temperature $T_{\mathrm{c}}$. In this model, fermions couple as a composite boson and condense to occupy a single bound state as the temperature drops. We derive a concise formula for $T_{\mathrm{c}}$ in terms of $\mathrm{\Delta }$ and ${\mathrm{\Delta }}^{\ast }$, namely: $$\frac{\mathrm{\Delta }}{k_{\mathrm{B}}{T}_{\mathrm{c}}}=1.4+4\log ({\mathrm{\Delta }}^{\ast }\mathrm{/}\mathrm{\Delta })\mathrm{.}$$ This expression reproduces the standard BCS gap-to-$T_{\mathrm{c}}$ ratio in the absence of a pseudogap, while naturally explaining its enhancement in unconventional superconductors. The model is supported by comparisons with experimental data from several cuprates and iron-based superconductors, which highlight its generality. This formulation also offers a theoretical explanation for the observed persistence of the pseudogap phase into the overdoped regime.