General theory

In several places in this book we have used the language and notions first developed by Landau to describe second-order phase transition, but which are used nowadays in a much broader context. Here we summarize the basics of this theory and illustrate different situations in which it is used. One can find a more detailed description for example in the brilliant original presentation of Landau and Lifshitz (1969), or in Khomskii (2010) (which is more or less followed below).

The original aim of Landau was to describe II order phase transitions – transitions in which a certain ordering, for example ferromagnetic, appears with decreasing temperature at some critical temperature Tc in a continuous manner. But it turned out later that the approach developed has much broader applicability than originally planned.

In thermodynamics and in statistical physics the optimal equilibrium state of a many-particle system is determined by the condition of the minimum of the Helmholtz freeenergy

F(V, T) = E − TS

or of the Gibbs free energy

Φ(P, T) = E − TS + PV

at given temperature and either fixed volume (C.1) or fixed pressure (C.2); more often in reality we are dealing with the second situation. When a certain ordering appears in the system – it may be magnetic ordering, for example ferro- or antiferromagnetic; or ferroelectricity; or an ordering in a structural phase transition – one can introduce a measure of such ordering, different for specific situations, which is called the order parameter; let us denote it η.