Lattice Magnetic Walks

TL;DR

This paper evaluates sums of walks for charged particles on a square lattice under magnetic fields, deriving expressions for propagators and Green's functions, with implications for understanding magnetic effects in lattice systems.

Contribution

It introduces a systematic method to include returning loops in lattice walk sums and derives explicit expressions for propagators at specific flux values, extending previous directed path results.

Findings

Derived expressions for propagators at commensurate flux values

Green's functions obtained for staggered flux configurations

Adding small returning loops preserves key features of directed path models

Abstract

Sums of walks for charged particles (e.g. Hofstadter electrons) on a square lattice in the presence of a magnetic field are evaluated. Returning loops are systematically added to directed paths to obtain the unrestricted propagators. Expressions are obtained for special values of the magnetic flux-per-plaquette commensurate with the flux quantum. For commensurate and incommensurate values of the flux, the addition of small returning loops does not affect the general features found earlier for directed paths. Lattice Green's functions are also obtained for staggered flux configurations encountered in models of high-Tc superconductors.