Heisenberg Groups in the Theory of the Lattice Peierls Electron: the Irrational Flux Case

TL;DR

This paper explores the quantum mechanics of electrons in magnetic fields and lattices using projective representation theory of Heisenberg groups, especially focusing on the irrational flux case and its relation to the Harper Hamiltonian.

Contribution

It provides a detailed group-theoretic analysis of the Landau and Peierls problems, highlighting the role of

Findings

The state space and Hamiltonian are derived from properties of the Heisenberg group.

For irrational flux, the groups are 'almost Heisenberg' with a distinguished irreducible representation.

The physics depends periodically on the flux/plaquette value.

Abstract

It is shown that the quantum mechanics of a charged particle moving in a uniform magnetic field in the plane (Landau) or on a planar lattice (Peierls) is described in all detail by the projective representation theory of the "euclidean" group of the appropriate configuration space. In the Landau case, a detailed description of the state space as well as the determination of the correct Hamiltonian follows from the properties of the real Heisenberg group, especially the fact that it has an essentially unique irreducible representation. In the Peierls case, the corresponding groups are infinite discrete translation groups centrally extended by the circle group. For irrational flux/plaquette (in units of the flux quantum) these groups are "almost Heisenberg" in the sense that they have a distinguished irreducible representation which plays, in the quantum theory, the role of the unique representation of the real Heisenberg group. The physics is fully determined by, and is periodic in, the value of the flux/plaquette. The Hamiltonian for nearest neighbour hopping is the Harper Hamiltonian. Vector potentials are not introduced.