ON A LINK BETWEEN NETWORK MODEL AND GROUP-THEORETICAL PERTURBATION TREATMENT IN THEORY OF DE HAAS-VAN ALPHEN EFFECT

In this paper we investigate from a more rigorous point of view the relation between the theory of the de Hass-van Alphen effect by Falicov and Stachowiak and the group-theoretical perturbation treatment in the case of magnetic breakdown. The two-dimensional Green's function of the Schrödinger equation, which plays an essential role in the theory of Falicov and Stachowiak, is calculated by using a well-known expansion of the evolution operator exp(-iHτ h {combining short stroke overlay}-1) in terms of successive orders of the lattice potential and by substituting rigorous expressions for the matrix elements, in the special case of a rectangular lattice. It is assumed that the magnetic flux η, in units |e| (hc)-1, passing through the unit cell of the lattice, is a rational number. Periodic boundary conditions are applied in order to avoid normalization difficulties. By using some assumptions, which are weaker than the assumptions in the theory of Falicov and Stachowiak, which is based on Pippard's network model, it is shown that the (integrated) two-dimensional Green's function is a linear combination of delta functions. The singularities in time correspond to periods of closed orbits on a network of coupled circles, as predicted by the theory of Falicov and Stachowiak in the case of a rectangular lattice. © 1969.