Electromagnetic lattice gauge invariance in two-dimensional discrete-time quantum walks

TL;DR

This paper develops a novel approach to incorporate electromagnetic gauge invariance into two-dimensional discrete-time quantum walks, aligning their behavior with principles of lattice gauge theories and recovering the Dirac equation in the continuum limit.

Contribution

It introduces a new method for embedding gauge fields in quantum walks that treats time and space symmetrically, enabling gauge-invariant, conserved currents and continuum Dirac dynamics.

Findings

Defined a gauge-invariant, conserved density current for quantum walks.

Demonstrated the continuum limit reproduces the Dirac equation.

Proposed a new framework for simulating electromagnetic interactions in quantum walks.

Abstract

Gauge invariance is one of the more important concepts in physics. We discuss this concept in connection with the unitary evolution of discrete-time quantum walks in one and two spatial dimensions, when they include the interaction with synthetic, external electromagnetic fields. One introduces this interaction as additional phases that play the role of gauge fields. Here, we present a way to incorporate those phases, which differs from previous works. Our proposal allows the discrete derivatives, that appear under a gauge transformation, to treat time and space on the same footing, in a way which is similar to standard lattice gauge theories. By considering two steps of the evolution, we define a density current which is gauge invariant and conserved. In the continuum limit, the dynamics of the particle, under a suitable choice of the parameters, becomes the Dirac equation, and the conserved current satisfies the corresponding conservation equation.