Gauge-invariant perturbation expansion in powers of electric charge for the density-of-states of a network model for charged-particle motion in a uniform background magnetic flux density

TL;DR

This paper develops a gauge-invariant perturbation expansion in powers of magnetic flux density for the density of states of a charged particle on a quantum network, providing explicit formulas up to third order.

Contribution

It introduces the first explicit gauge-invariant expansion for the density of states in a magnetic flux, valid up to third order, for particles on quantum networks.

Findings

Explicit gauge-invariant formulas up to third order in flux density.

Valid for both global and local traces of the resolvent.

New expressions not previously published.

Abstract

An explicitly-gauge-invariant expansion in powers of $e/\hbar$ times the magnetic flux density is formally obtained for the density of states (as characterized by the trace of the resolvent $\widehat G$ = $(\omega - \hat h)^{-1}$) of a charged particle moving on a Hermitian quantum network that is embedded in a Euclidean background that supports a uniform magnetic flux density. The explicit expressions, given here up to third order in the flux density, are also valid for the ``local trace'' (the trace of $\widehat P_i \widehat G$, where $\widehat P_i$ is the projector on a network node), and do not appear to have been previously given.