A new method to building Dirac quantum walks coupled to electromagnetic fields

TL;DR

This paper introduces a systematic method for constructing Dirac Quantum Walks coupled to electromagnetic fields, demonstrating its effectiveness through multiple lattice examples and exploring limitations on certain lattice structures.

Contribution

A new systematic method for building Dirac Quantum Walks coupled to electromagnetic fields, applied to various lattice structures and analyzing their feasibility.

Findings

Successfully derived EM coupling for 3D cubic lattice walk

Re-derived DQWs on triangular and honeycomb lattices, showing they are unique with spinors on vertices

Identified lattices like rhombohedral that cannot support DQWs

Abstract

A quantum walk whose continuous limit coincides with Dirac equation is usually called a Dirac Quantum Walk (DQW). A new systematic method to build DQWs coupled to electromagnetic (EM) fields is introduced and put to test on several examples of increasing difficulty. It is first used to derive the EM coupling of a well-known $3D$ walk on the cubic lattice. Recently introduced DQWs on the triangular and honeycomb lattice are then re-derived, showing for the first time that these are the only DQWs that can be defined with spinors living on the vertices of these lattices. As a third example of the method's effectiveness, a new $3D$ walk on a parallelepiped lattice is derived. As a fourth, negative example, it is shown that certain lattices like the rhombohedral lattice cannot be used to build DQWs. The effect of changing representation in the Dirac equation is also discussed.