Lifshitz’s Criterion for Symmetry Changes in Second-Order Phase Transitions

Abstract

Some points implicit in the Landau theory of second-order phase transitions are clarified. Lifshitz’s criterion is then directly derived by requiring the absence of k-linear terms A_k(p, T), the second-order coefficient in Landau’s expansion of free energy. It is shown that the presence of k-linear terms leads to invariants bilinear in the order parameter and its spatial derivative as proposed by Lifshitz. The theory is developed in parallel with the shallow impurity problems in semiconductors and the k·p perturbation theory. Lifshitz’s criterion is shown to be equivalent to the absence of k-linear terms in energy bands. The equivalence makes the application of Lifshitz’s criterion easier