In a superconductor, the complex order parameter ψ that describes the condensation of Cooper pairs becomes finite, and the U(1) symmetry of rotating its phase is spontaneously broken. Tab ① draws the “Mexican hat” of the Ginzburg–Landau (GL) free energy f = a|ψ|² + (b/2)|ψ|⁴ + |(−iħ∇ − e*A)ψ|²/2m*, and shows how raising the temperature (a = a₀(T − Tc) becoming positive) destabilizes the symmetry-broken state and turns the hat into a bowl, as you vary the coefficients a and b, the effective mass m*, the effective charge e*, and the vector potential |A|, using coefficients determined from the measured values of Al, Ga, In, Tl, Sn, Pb, Zn, Cd, Hg, V, Nb, and Ta. Tab ② solves the BCS mean-field gap equation to obtain the transition temperature Tc and the gap Δ(T) (with the McMillan formula and the Gor’kov correspondence). Tab ③ examines the two oscillations on the hat — the Higgs mode of the amplitude and the Nambu–Goldstone mode of the phase — and the Anderson–Higgs mechanism, and tab ④ shows that the same free energy appears as a soft mode of the lattice (the “phonons with imaginary frequencies” found in DFT). This follows Chapters 2 and 3 of Naoto Tsuji’s “Physics of Nonequilibrium Superconductivity” (in Japanese; Summer School for Young Researchers in Condensed Matter Physics). For the equations, derivations, and literature survey, see the explanation page (in Japanese).
Companion page: Explanation — GL theory, BCS theory, spontaneous symmetry breaking, and the Higgs/Goldstone modes of phonons (superconductivity-derivation.html, in Japanese) | Related: Cooper Pair Simulator (lattice wake and the BCS–BEC crossover), Relativistic Quantum Mechanics Simulator, Elementary Particle Simulator | Mochizuki Research Group, Tokyo University of Science