Quantum Walks and discrete Gauge Theories

TL;DR

This paper demonstrates that quantum walks can simulate discrete gauge theories, introducing a new family of 2D walks with gauge invariance and electromagnetic interactions, bridging discrete models with continuous gauge field dynamics.

Contribution

The authors introduce a new family of 2D discrete-time quantum walks that exhibit exact discrete gauge invariance and simulate electromagnetic interactions, including a discrete Maxwell's equations framework.

Findings

Quantum walks can simulate Dirac fermions coupled to electromagnetic fields.

The walks exhibit exact discrete local U(1) gauge invariance.

Numerical simulations show phenomena like Bloch oscillations and localization.

Abstract

A particular example is produced to prove that quantum walks can be used to simulate full-fledged discrete gauge theories. A new family of $2D$ walks is introduced and its continuous limit is shown to coincide with the dynamics of a Dirac fermion coupled to arbitrary electromagnetic fields. The electromagnetic interpretation is extended beyond the continuous limit by proving that these DTQWs exhibit an exact discrete local $U(1)$ gauge invariance and possess a discrete gauge-invariant conserved current. A discrete gauge-invariant electromagnetic field is also constructed and that field is coupled to the conserved current by a discrete generalization of Maxwell equations. The dynamics of the DTQWs under crossed electric and magnetic fields is finally explored outside the continuous limit by numerical simulations. Bloch oscillations and the so-called ${\bf E} \times {\bf B}$ drift are recovered in the weak-field limit. Localization is observed for some values of the gauge fields.