Field Theory on the von Neumann Lattice and the Quantized Hall Conductance of Bloch Electrons

TL;DR

This paper develops a field-theoretical framework for quantum Hall systems on the von Neumann lattice, generalizing topological formulas for Hall conductance to Bloch electrons and clarifying the relation between winding and Chern numbers.

Contribution

It introduces a new set of one-particle states based on the von Neumann lattice and extends topological Hall conductance formulas to Bloch electrons.

Findings

Established a formalism connecting winding number and Chern number.

Generalized the topological Hall conductance formula for Bloch electrons.

Clarified the relation between topological invariants in quantum Hall systems.

Abstract

We construct useful sets of one-particle states in the quantum Hall system based on the von Neumann lattice. Using the set of momentum states, we develop a field-theoretical formalism and apply the formalism to the system subjected to a periodic potential. The topological formula of the Hall conductance written by the winding number of propagator is generalized to Bloch electrons. The relation between the winding number and the Chern number is clarified.