Critical magnetic fields H_c(x), H_c2(x) and flux φ(x) vs superfluid density x in a superconductor

By the BCS framework [Bardeen, Cooper and Schrieffer, Phys. Rev. 108, 1175 (1957); Nam, Phys. Rev. 156, 470 (1967)], with the notion of finite pairing interaction energy range [the cut-off energy] T_d=2zT_c [critical temperature], we present novel results, at zero temperature, H_c(x)= b(x)H_c(1), H_c(1) [BCS]= [4πN(0)]^1/2Δ(1), b(x)= 2Δ(x)/Δ(1)(1+x), H_c2(x)= b²(x)H_c2(1), H_c2(1) 2πξ²(1) [Ginzburg-Landau coherence length square] =φ(1)[flux quantum]=[hc/2e], ξ(1)=ξ(x)b(x) = πξ_BCS/12^½, with the superfluid density x=ρ_s/ρ= tanh(G/2) [Nam, APSMAR22/A61.4 (2022)], G= 1/N(0)V =∫₀^z dy tanh(y)/y = Σ_j (2/w_j) tan^-1(z/w_j), w_j=(π/2) j(odd integer) [Nam, Phys. Lett. A193, 111 (1994); (E) ibid A197, 458 (1995)], Δ(x)/2T_c =A(z)= z/sinhG, where N(0) and V are the density of states/spin at a reference energy and the BCS pairing interaction energy, respectively. A(∞) [BCS]= 0.88 and A(+0)=1. The upper value H_c2(x) is to be 5.165 H_c2(1). The different values of local ξ(x)= ξ(1)/b(x) may account for the fractional flux quanta reported by Iguchi et al., Science 380, 1244 (2023). Superconductors [x<1] are multiconnected [Nam, Phys. Lett. A198, 447 (1995)].