
Phase transitions related to superconductivity and electronic order in materials.A superconducting phase transition is the abrupt change in a material's electronic state when it begins to conduct electricity with zero resistance. This transition occurs at a critical temperature, below which electrons form pairs and move coherently without scattering. It is a macroscopic quantum phenomenon, meaning quantum effects become visible at a scale much larger than atoms, and it underpins technologies like MRI machines and particle accelerators.
Heike Kamerlingh Onnes discovered superconductivity in 1911 when he observed mercury's resistance drop to zero at 4.2 K. The theoretical explanation came in 1957 with the BCS theory by Bardeen, Cooper, and Schrieffer.
In a normal metal, electrons move independently and collide with the lattice, causing resistance. Below the critical temperature, electrons bind into Cooper pairs via lattice vibrations (phonons). These pairs condense into a single quantum state described by a single wavefunction, allowing them to flow without energy loss. The transition is a phase transition because the material's symmetry changes: the electromagnetic gauge symmetry is spontaneously broken, giving rise to the Meissner effect (expulsion of magnetic fields).
Superconducting phase transitions are used in MRI magnets, particle accelerators like the LHC, and quantum computing qubits (e.g., in superconducting films for magnetic vortex qubits).
The transition temperature can be tuned by stretching a carbon film, as shown in recent experiments, making superconductivity controllable by mechanical strain.
| F | variable | Landau free energy density — free energy density joule per cubic metre |
| ψ | variable | order parameter — order parameter |
| a | variable | coefficient of quadratic term, temperature-dependent — landau coefficient a joule per cubic metre |
| b | variable | coefficient of quartic term — landau coefficient b joule per cubic metre |
| c | variable | coefficient of sixth-order term — landau coefficient c joule per cubic metre |
| f_ext | variable | external field coupled to the order parameter — external field joule per cubic metre |
| \Delta(\mathbf{k}) | variable | Superconducting gap as a function of wavevector — superconducting gap joule |
| k_x | variable | Wavevector component along x-axis — wavevector component reciprocal metre |
| k_y | variable | Wavevector component along y-axis — wavevector component reciprocal metre |
| \Delta Q | variable | Accumulated charge change in the superconductor — electric charge coulomb |
| \Delta \Phi | variable | Change in magnetic flux through the superconducting loop — magnetic flux weber |
| \nu | variable | Filling factor (integer or fractional) of the quantum Hall state — dimensionless |
| e | constant | Magnitude of the electric charge of a single electron or proton — elementary charge = 1.602176634e-19 coulomb |
| \Phi_0 | constant | Superconducting flux quantum, h/(2e), the fundamental unit of magnetic flux in a superconductor — magnetic flux quantum = 2.067833848e-15 weber |
| B(E1) | variable | Electric dipole reduced transition probability — reduced transition probability coulomb squared metre squared |
| Z | variable | Atomic number (number of protons) — atomic number |
| e | variable | Elementary charge — elementary charge coulomb |
| ⟨r²⟩ | variable | Mean square radius of the charge distribution — mean square radius metre squared |
| π | constant | Ratio of a circle's circumference to its diameter — pi = 3.14159 |
| E | variable | quasiparticle energy — energy joule |
| ε | variable | single-particle energy relative to chemical potential — energy joule |
| μ | variable | chemical potential — chemical potential joule |
| Δ | variable | superconducting gap parameter — energy gap joule |


















