A Phenomenological Organization of BCS Superconductivity via Fermion–Boson Duality: From an Occupation-Probability Decomposition to the Pseudogap and the BCS–BEC Crossover

The starting point of this paper is the Mühlschlegel formula Δ(T)/Δ0 ≃ tanh{1.74·√(T_c/T − 1)}, which closely approximates the weak-coupling BCS gap. By an elementary identity, the tanh can be written exactly as the difference of two complementary logistic occupation probabilities. Hence, within the range of this approximation, the BCS gap can be read as the difference L_B^(e) − L_F^(e) between the B-type and F-type occupation probabilities of the electron sector, where the B-type component behaves bosonically, like a Cooper pair, and the F-type component is the ordinary quasiparticle-like component. Taking this mathematical fact as a core, we extend the phenomenology of Fermion–Boson Duality (FBD). In addition to the electron sector we introduce a gauge sector, representing superconductivity through four occupation probabilities {L_B^(e), L_F^(e), L_B^(γ), L_F^(γ)}. When the transition centers of the two sectors coincide, weak-coupling BCS is recovered; when they separate, a staircase structure 1 → 1/2 → 0 appears in the observed gap, a candidate for the pseudogap of high-temperature superconductivity. When the tanh arguments of the two sectors have opposite signs, a competing structure represents the BCS–BEC crossover of cold-atom systems as a mixture of BCS-like and BEC-like occupation probabilities, and the Bertsch parameter at unitarity is interpreted as the admixture fraction of BCS character. What is rigorous here is the decomposition of the Mühlschlegel tanh into an occupation-probability difference. The reduction to four degrees of freedom, the gap averaging, the Matsubara connection, and the BCS–BEC interpolation are phenomenological hypotheses to be tested against experiment.