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 $\Delta$, the pseudogap $\Delta^\ast$, 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 $\Delta$ and $\Delta^\ast$, namely: $$\frac{\Delta}{k_\mathrm{B} T_\mathrm{c}} = 1.4+4\log(\Delta^\ast/\Delta).$$ 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.