Dots and boxes algorithm for Peierls substitution: application to multidomain topological insulators

Magnetic fields can be introduced into discrete models of quantum systems by the Peierls substitution. For tight-binding Hamiltonians, the substitution results in a set of (Peierls) phases that are usually calculated from the magnetic vector potential. As the potential is not unique, a convenient gauge can be chosen to fit the geometry and simplify calculations. However, if the magnetic field is non-uniform, finding a convenient gauge is challenging. In this work we propose to bypass the vector potential determination by calculating the Peierls phases exclusively from the gauge-invariant magnetic flux. The phases can be assigned following a graphic algorithm reminiscent of the paper and pencil game 'dots and boxes'. We showcase the method implementation by calculating the interference phenomenon in a modified Aharonov-Bohm ring and propose a phase assignation alternative to the Landau gauge to reproduce the Half Integer Quantum Hall Effect in graphene. A non-uniform magnetic field case is addressed by considering a multi-domain Chern insulator to study the effects of domain walls in resistance and current quantization. It is found that adding decoherence and a finite temperature into the model results in quantized resistances that are in good agreement with experiments made with multi-domain intrinsic topological insulators.