Conserved and broken symmetries in the renormalization of the phase space lattice Hamiltonian

Abstract

The phase space lattice Hamiltonian is a realistic model for Bloch electrons in a magnetic field. It has a fractal spectrum when the lattice has centres of threefold or fourfold rotational symmetry. This fact has been explained using a renormalization group (RG) method, assuming that the RG transformation preserves the symmetry of the Hamiltonian. The symmetry preservation property has previously been demonstrated for fourfold rotation; the threefold case is considerably more difficult to analyse. In this paper we present a simplified form of the RG equations which clearly exhibits the threefold symmetry preservation. We also discuss the case of sixfold rotational symmetry, for which the symmetry of the Hamiltonian may be reduced to threefold under the action of the RG.