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

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.