Fermion Doubling in Dirac Quantum Walks

TL;DR

This paper investigates fermion doubling in quantum walk models simulating Dirac particles, identifies issues caused by doublers and pseudo-doublers, and proposes a new family of walks that mitigate these problems while maintaining the correct continuum limit.

Contribution

It introduces a family of quantum walks free of fermion doubling and pseudo-doubling, improving the simulation of Dirac particles in discrete spacetime models.

Findings

Proposed quantum walks eliminate fermion doublers and pseudo-doublers.

Identified issues with spurious solutions and vacuum stability.

Modified walks allow non-zero stay probability, reducing unwanted solutions.

Abstract

We consider discrete spacetime models known as quantum walks, which can be used to simulate Dirac particles. In particular we look at fermion doubling in these models, in which high momentum states yield additional low energy solutions which behave like Dirac particles. The presence of doublers carries over to the `second quantised' version of the walks represented by quantum cellular automata, which may lead to spurious solutions when introducing interactions. Moreover, we also consider pseudo-doublers, which have high energy but behave like low energy Dirac particles, and cause potential problems regarding the stability of the vacuum. To address these issues, we propose a family of quantum walks, that are free of these doublers and pseudo-doublers, but still simulate the Dirac equation in the continuum limit. However, there remain a small number of additional low energy solutions which do not directly correspond to Dirac particles. While the conventional Dirac walk always has a zero probability for the walker staying at the same point, we obtain the family of walks by allowing this probability to be non-zero.