An exact renormalisation group for Bloch electrons in a magnetic field

Abstract

Bloch electrons in a magnetic field can be modelled by a one-dimensional Hamiltonian H=H(x, p), which is periodic in x and p. When beta , the ratio of Planck's constant h to the area of a unit cell is a rational number p/q, the spectrum is a set of q Bloch bands. If beta is perturbed away from this rational value, the spectrum becomes a Cantor set, but it is still possible to define a set of generalised Bloch waves which form a complete basis set for a given band. Projecting these states into the band, and taking matrix elements of the Hamiltonian leads to a new effective Hamiltonian H1v(x,p) and Planck constant h'v describing the spectrum of the vth band. Since degrees of freedom have been eliminated by projecting into a band, this is a renormalisation group (RG) transformation. The RG transformation can be iterated indefinitely if the measure of the spectrum is zero or if beta is a Liouville number. The measure of the spectrum vanishes if H(x,p) has centres of threefold or fourfold symmetry in the phase plane, and the RG transformation explains the hierarchical structure of the spectrum observed in these cases.