Bloch electrons in a magnetic field: Topology of quantum states and transport

Quantum states and Hall conductances of electrons in $n$-type heterojunctions and holes in $p$-type heterojunctions in a field of a lateral superlattice and a perpendicular magnetic field were studied. It is shown that the energy spectrum of magnetic subbands in a periodic potential without inversion center is not symmetric about the reversal of the quasi-momentum sign. The properties of wave functions and the related topological invariants determining the Hall conductance were examined. The method of calculating the magnetic Bloch states of holes was developed on the basis of the Luttinger Hamiltonian, allowing the spin and spin-orbit interactions to be taken into account in this problem. The Hall conductance quantization law was determined for 2D holes in a periodic superlattice potential.