By I. T. Todorov, D. Ter Haar

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26), a differential equation – the famous Ginzburg–Landau equation. 4 Inhomogeneous situations Φ 17 G>0 G<0 ∇h Fig. 11 written as 1 2e −i − A h∇ − 2m c 2 ψ + Aψ + 2B|ψ|2 ψ = 0 . 27) has a form similar to the Schr¨odinger equation, but with a nonlinear term ∼ψ 3 . Close to Tc , when the order parameter ψ → 0, this equation can be linearized, and then it is indeed exactly equivalent to the corresponding Schr¨odinger equation. This analogy is very useful for the treatment of many problems in superconductivity, such as the upper critical magnetic field Hc2 , the formation and properties of vortices, etc.

4) determines phonon frequencies, in the anharmonic case depends on x). 49) ω dV d ln V which is called the Gr¨uneisen approximation; γ is the Gr¨uneisen constant (usually, in ordinary crystals, γ ∼ 1–2). The total free energy as a function of volume can then be written as 1 F (V ) = 2κ δV V 2 +T ln 2 sinh q ωq (V ) 2T . 50) is the elastic energy after deformation δV , and κ is the lattice compressibility (inverse bulk modulus). 50) we considered the situation when we (artificially) fix the volume of the system V , which may differ from the equilibrium volume without phonons by the distortion δV .

G. by application of pressure, a magnetic field or some other control parameter g, see Fig. 14 (the grey region is here the region of classical fluctuations). In this case, if Tc tends to zero, quantum effects start to play a more and more important role. g. pressure P > Pc ) 24 General theory of phase transitions T Tc ordered phase c QCP Fig. 14 may be disordered even at T = 0, not because of classical, but because of quantum fluctuations – we can speak of a quantum disordered phase. g. magnetic ordering may be suppressed for P > Pc ), but it is a unique quantum state described by a, maybe very complicated, but unique wavefunction.